Lecture Notes in Bioinformatics
5688
Edited by S. Istrail, P. Pevzner, and M. Waterman Editorial Board: A. Apostolico S. Brunak M. Gelfand T. Lengauer S. Miyano G. Myers M.F. Sagot D. Sankoff R. Shamir T. Speed M. Vingron W. Wong
Subseries of Lecture Notes in Computer Science
Pierpaolo Degano Roberto Gorrieri (Eds.)
Computational Methods in Systems Biology 7th International Conference, CMSB 2009 Bologna, Italy, August 31 September 1, 2009 Proceedings
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Series Editors Sorin Istrail, Brown University, Providence, RI, USA Pavel Pevzner, University of California, San Diego, CA, USA Michael Waterman, University of Southern California, Los Angeles, CA, USA Volume Editors Pierpaolo Degano Università di Pisa, Dipartimento di Informatica Largo Bruno Pontecorvo, 3, 56127, Pisa, Italy Email:
[email protected] Roberto Gorrieri Università degli Studi di Bologna Dipartimento di Scienze dell’Informazione Mura Anteo Zamboni 7, 40127 Bologna, Italy Email:
[email protected] Library of Congress Control Number: 2009932359
CR Subject Classification (1998): J.3, I.6, F.1.1, F.4.1, G.2.2 LNCS Sublibrary: SL 8 – Bioinformatics ISSN ISBN10 ISBN13
03029743 3642038441 Springer Berlin Heidelberg New York 9783642038440 Springer Berlin Heidelberg New York
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Preface
This volume contains the proceedings of the 7th Conference on Computational Methods in Systems Biology (CMSB 2009), held in Bologna, from August 31 to September 1, 2009. The ﬁrst CMSB was held in Trento in 2003, bringing together life scientists, computer scientists, engineers and physicists. The goal was to promote the convergence of diﬀerent disciplines aiming at a new understanding and description of biological systems, ﬁrmly ground in formal models, supported by computational languages and tools, and oﬀering new methods of analysis. The conference then moved to Paris in 2004, Edinburgh in 2005, Trento in 2006, Edinburgh in 2007 and Rostock/Warnem¨ unde in 2008. This year the conference attracted about 45 submissions form 18 countries, mainly from Europe and North America, but also from Asia and Australia. We wish to thank all authors for their interest in CMSB 2009. After careful discussions, the Programme Committee eventually selected 18 papers for presentation at the conference. Each of them was accurately refereed by at least three reviewers, who delivered detailed and insightful comments and suggestions. The Conference Chairmen warmly thank all the members of the Programme Committee and all their subreferees for the excellent support they gave, as well as for the friendly and constructive discussions. We also would like to thank the authors for having revised their papers to address the comments and suggestions by the referees. This year we also had a poster session, hosting ten short presentations, each refereed by at least two members of the Programme Committee. We would like to thank their authors for the bright presentations of their workinprogress. The abstracts of these posters are collected in a separate Technical Report, available at http://compass2.di.unipi.it/TR/Files/TR0909.pdf.gz. The conference programme was enriched by the outstanding invited talks of Rita Casadio, John K. Heath and Corrado Priami, whom we warmly thank. Their contributions are also included here. The conference this year was jointly organized with CMSB 2009, emphasizing the close connections and similarities between concurrent, artiﬁcial systems, and biological, natural systems. The joint invited talk by Corrado Priami further illustrated this correspondence. We would like to thank all the people who contributed to the organization of CMSB 2009, and the generous support from the Alma Mater Studiorum – Universit`a degli Studi di Bologna and from Microsoft Research Cambridge. We are also grateful to Andrei Voronkov, who allowed us to use the wonderful free conference software system EasyChair, which we used for the electronic submission of papers, the refereeing process and the Programme Committee work. Pierpaolo Degano Roberto Gorrieri
Organization
Steering Committee Finn Drabløs Fran¸cois Fages David Harel Monika Heiner Michael H¨ornquist Satoru Miyano Gordon Plotkin Corrado Priami Adelinde M. Uhrmacher
Norwegian University of Science and Technology, Trondheim, Norway INRIA Rocquencourt, France Weizmann Institute of Science, Israel TU Cottbus, Germany Link¨ oping University, Sweden University of Tokyo, Japan University of Edinburgh, UK MSR  University of Trento Centre for Comp. and Systems Biology, Italy University of Rostock, Germany
Programme Committee Pierpaolo Degano Finn Drabløs Fran¸cois Fages Stephen Gilmore Roberto Gorrieri Monika Heiner Adaoha Elizabeth C. Ihekwaba Marta Kwiatkowska Pietro Li` o Satoru Miyano Mark van Rossum Grzegorz Rozenberg Carolyn Talcott Adelinde M. Uhrmacher
Universit` a di Pisa, Italy (Cochair) Norwegian University of Science and Technology, Norway INRIA Rocquencourt, France University of Edinburgh, UK Universit` a di Bologna, Italy (Cochair) TU Cottbus, Germany CoSBI, Italy University of Oxford, UK Computing Lab Cambridge, UK University of Tokyo, Japan University of Edinburgh, UK Leiden University, The Netherlands Stanford Research Institute, USA University of Rostock, Germany
Organizing Committee Cinzia Di Giusto Roberto Gorrieri Cristian Versari Antonio Vitale Gianluigi Zavattaro
Universit`a Universit` a Universit` a Universit`a Universit` a
di di di di di
Bologna, Bologna, Bologna, Bologna, Bologna,
Italy Italy Italy (Chair) Italy Italy
VIII
Organization
Additional Reviewers Marco Aldinucci Gr´egory Batt Chiara Bodei Luca Bortolussi Andrea Bracciali Mario Bravetti Matteo Cavaliere Steven Eker Vashti Galpin Cinzia Di Giusto Maria Luisa Guerriero Russ Harmer Jane Hillston Espen Højsgaard Mathias John
Roberto Larcher Chen Li Fei Liu Laurence Loewe Vittorio Maniezzo Roberto Marangoni Radu Mardare Carsten Maus Richard Mayr Emanuela Merelli Paolo Milazzo Ivan Mura Masao Nagasaki Gethin Norman Dave Parker
Carla Piazza Alberto Policriti Davide Prandi Christian Rohr Martin Schwarick Teppei Shimamura Heike Siebert Joris Slegers Sylvain Soliman Ashish Tiwari Cristian Versari Antonio Vitale
Table of Contents
Prediction of ProteinProtein Interacting Sites: How to Bridge Molecular Events to Large Scale Protein Interaction Networks (Invited Talk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lisa Bartoli, Pier Luigi Martelli, Ivan Rossi, Piero Fariselli, and Rita Casadio
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The Equivalence between Biology and Computation (Invited Talk) . . . . . John K. Heath
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BlenX4Bio – BlenX for Biologists (Invited Talk) . . . . . . . . . . . . . . . . . . . . . . Corrado Priami, Paolo Ballarini, and Paola Quaglia
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Modelling Biological Clocks with BioPEPA: Stochasticity and Robustness for the Neurospora Crassa Circadian Network . . . . . . . . . . . . . Ozgur E. Akman, Federica Ciocchetta, Andrea Degasperi, and Maria Luisa Guerriero Quantitative Pathway Logic for Computational Biology . . . . . . . . . . . . . . . Michele Baggi, Demis Ballis, and Moreno Falaschi A PrizeCollecting Steiner Tree Approach for Transduction Network Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marc BaillyBechet, Alfredo Braunstein, and Riccardo Zecchina Formal Analysis of the Genetic Toggle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giampaolo Bella and Pietro Li` o Control Strategies for the Regulation of the Eukaryotic Heat Shock Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elena Czeizler, Eugen Czeizler, RalphJohan Back, and Ion Petre Computing Reachable States for Nonlinear Biological Models . . . . . . . . . . Thao Dang, Colas Le Guernic, and Oded Maler On Coupling Models Using ModelChecking: Eﬀects of Irinotecan Injections on the Mammalian Cell Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elisabetta De Maria, Fran¸cois Fages, and Sylvain Soliman The κLattice: Decidability Boundaries for Qualitative Analysis in Biological Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giorgio Delzanno, Cinzia Di Giusto, Maurizio Gabbrielli, Cosimo Laneve, and Gianluigi Zavattaro
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68
83
96
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X
Table of Contents
Approximation of Event Probabilities in Noisy Cellular Processes . . . . . . Fr´ed´eric Didier, Thomas A. Henzinger, Maria Mateescu, and Verena Wolf
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Equivalence and Discretisation in BioPEPA . . . . . . . . . . . . . . . . . . . . . . . . . Vashti Galpin and Jane Hillston
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Improved Parameter Estimation for Completely Observed Ordinary Diﬀerential Equations with Application to Biological Systems . . . . . . . . . . Peter Gennemark and Dag Wedelin
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A Bayesian Approach to Model Checking Biological Systems . . . . . . . . . . Sumit K. Jha, Edmund M. Clarke, Christopher J. Langmead, Axel Legay, Andr´e Platzer, and Paolo Zuliani
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Dynamic Compartments in the Imperative πCalculus . . . . . . . . . . . . . . . . Mathias John, C´edric Lhoussaine, and Joachim Niehren
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Probabilistic Approximations of Signaling Pathway Dynamics . . . . . . . . . Bing Liu, P.S. Thiagarajan, and David Hsu
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A Reduction of Logical Regulatory Graphs Preserving Essential Dynamical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aur´elien Naldi, Elisabeth Remy, Denis Thieﬀry, and Claudine Chaouiya On the Use of Stochastic Petri Nets in the Analysis of Signal Transduction Pathways for Angiogenesis Process . . . . . . . . . . . . . . . . . . . . . Lucia Napione, Daniele Manini, Francesca Cordero, Andr´ as Horv´ ath, Andrea Picco, Massimiliano De Pierro, Simona Pavan, Matteo Sereno, Andrea Veglio, Federico Bussolino, and Gianfranco Balbo
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CSL Model Checking of Biochemical Networks with Interval Decision Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Martin Schwarick and Monika Heiner
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Qualitative Transition Systems for the Abstraction and Comparison of Transient Behavior in Parametrized Dynamic Models . . . . . . . . . . . . . . . . . Hayssam Soueidan, Gr´egoire Sutre, and Macha Nikolski
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Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Prediction of ProteinProtein Interacting Sites: How to Bridge Molecular Events to Large Scale Protein Interaction Networks Lisa Bartoli1, Pier Luigi Martelli1, Ivan Rossi2, Piero Fariselli1, and Rita Casadio1 1
Biocomputing Group, University of Bologna, CIRB/Department of Biology, Via San Giacomo 9/2, Bologna, Italy 2 BioDec s.r.l., Casalecchio di Reno, Bologna, Italy {casadio,lisa,gigi,ivan,piero}@biocomp.unibo.it http://www.biocomp.it
Abstract. Most of the cellular functions are the result of the concerted action of protein complexes forming pathways and networks. For this reason, efforts were devoted to the study of proteinprotein interactions. Largescale experiments on whole genomes allowed the identification of interacting protein pairs. However residues involved in the interaction are generally not known and the majority of the interactions still lack a structural characterization. A crucial step towards the deciphering of the interaction mechanism of proteins is the recognition of their interacting surfaces, particularly in those structures for which also the most recent interaction network resources do not contain information. To this purpose, we developed a neural networkbased method that is able to characterize protein complexes, by predicting amino acid residues that mediate the interactions. All the Protein Data Bank (PDB) chains, both in the unbound and in the complexed form, are predicted and the results are stored in a database of interaction surfaces (http://gpcr.biocomp.unibo.it/zenpatches). Finally, we performed a survey on the different computational methods for proteinprotein interaction prediction and on their training/testing sets in order to highlight the most informative properties of protein interfaces. Keywords: proteinprotein interaction site, neural network, patch smoothing, interacting surfaces.
1 Introduction The analysis of proteinprotein interaction surfaces has a long history, dating back to the seventies. Chotia and Janin [1] first analyzed a small number of structures to highlight some principles of proteinprotein recognition. Much later Thornton and coworkers [2] focused on the features of patches of interacting residues within homodimers, showing how features as solvation potential, residue interface propensity, hydrophobicity, planarity, protrusion and accessible surface area were good candidates for the description of protein interfaces. With the increasing number of proteins known with atomic resolution in the Protein Data Bank (PDB, http://www.rcsb.org/ pdb/ home/home.do), an increasing number of efforts were devoted to the issue of P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 1–17, 2009. © SpringerVerlag Berlin Heidelberg 2009
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extracting basic features of interacting protein complexes with the aim of predicting possible proteinprotein interactions [for recent reviews see 35]. This became particularly urgent when it was recognized that proteinprotein interactions were at the basis of a new paradigm of interpreting metabolic pathways. Indeed, interactions between proteins are the core of almost all the biochemical processes that mediate cellular function [6]. Largescale experiments on whole genomes allowed the identification of many interacting protein pairs but the residues involved in these interactions are generally not known and the vast majority of the interactions remain to be structurally characterized. The structural characterization of proteinprotein complexes remains an expensive and time consuming process and it is particularly problematic for transient complexes (see below). Molecular recognition in protein complexes can be predicted with docking techniques, based on the notions of surface complementarity and electrostatics rules, and fitting together the surfaces of two or more known structures. Significant advances have been achieved in this field: however, the molecular forces involved in the complex association process are not completely understood and the methods are also influenced by the conformational changes that often take place upon proteinprotein binding [7]. A possible alternative for characterizing protein complexes is a thorough study of protein complexes known with atomic resolution and the implementation of methods capable of inferring from known examples the general rules for predicting the propensity of the amino acid residues to mediate proteinprotein interactions. It is a common believe that residues present at the protein interfaces should be easier to predict provided that its distinguishing features are known [8]. With this underlying assumption many studies have attempted to characterise the residues in proteinprotein interfaces and from this to highlight all those rules that may help in the development of predictive methods. In the following we will describe features of interacting protein complexes as derived from structural data bases, thestate of the art methods that have been differently trained on these features and our approach. We will also describe the results of some experiments trying to emphasize the role of data bases in getting the evaluation scores of the predictive methods.
2 General Features of Interacting Protein Complexes Earlier works were restricted to a limited subset of oligomeric proteins in the PDB while more recent works have been able to extend the analysis to a larger dataset of protein structures that became available in the most recent PDB releases. When in vitro studies are also available, one possibility is to take into consideration the strength of the interaction, whether the interaction is transient or not and whether or not the complexes are formed by copies of the same chain. These later studies suggest that the composition of interacting residues in the interfaces is different in each subset: for example homodimers tend to have more hydrophobic residues in their interfaces than heterodimers. It has also been observed that small interfaces tend to have a higher content of charged and polar residues if compared to larger ones. A further distinction can be made between obligate, nonobligate, transient and permanent complexes. Obligate interactions are those where the monomers do not
Prediction of ProteinProtein Interacting Sites
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exist as stable structures in vivo while in nonobligate interactions the monomers can exist independently of the complex. The distinction between transient and permanent interactions was originally based on the lifetime of the association. A permanent interaction is quite stable and for this reason, in general, it exists only in its complexed form, while a transient interaction associates and dissociates in vivo. Structural and functional obligate interactions usually are routinely permanent while nonobligate interactions can be transient or permanent. The interfaces of obligate complexes have clusters of hydrophobic residues while the transient interfaces have a significant number of polar residues that contribute to electrostatic interactions [9]. Together this findings suggest that transient interfaces are in general smaller with respect to obligate ones. Recently, it has been observed that in obligate complexes, monomers share more molecular functions (following the GO annotation) than in transient complexes. Moreover, residues in the obligate interfaces tend to evolve slower than residues in the transient ones. This indicates that obligate complexes residues tend to coevolve with the interacting partner while in transient complexes a higher rate of substitution leaves no marked signature of correlated mutations [for more references see 36]. Different contact patterns between obligate and nonobligate interfaces have also been detailed. It was found that contacts among obligate complexes are mainly non polar and that protein chains belonging to obligate complexes show a higher number of contacts per interface than nonobligate complex chains. Unfortunately most interactions do not readily fall into a definite class and for this reason classifying complexes by these definitions is not so simple [6]. Furthermore, secondary structure elements seem to give a major contribute to the formation of the interface in obligate complexes. The interaction between secondary structure motifs is driven by the interaction between main chain atoms, especially in the formation of obligate interfaces. Finally, βsheet formation across the interacting units was observed only in obligate interactions. Peculiar characteristics of interaction sites can be captured by positionspecific sequence profiles such as those generated from PSIBLAST multiplesequence alignments [811]. Residue conservation at the interface is observed to be slightly higher than those of general surface residues, although it is not significantly different from those in the protein interior. The discriminatory power of the evolutionary conservation has been observed to be stronger for obligate and more permanent interactions [5]. Summing up, several different chemicophysical and topological features can be associated to interacting protein surfaces. However, correlations between these features and proteinprotein binding sites are so subtle that they cannot be predicted with linear models alone.
3 Prediction of Interacting Residues: TheStateoftheArt Methods In spite of the differences detailed above, protein interfaces are not endowed with features that make them simple to be computed from specific sets of rules. For this reason computational methods that attempt to identify interface residues are very valuable. The most widely adopted machine learning methods, Neural Networks (NN)
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and Support Vector Machines (SVM), exploit the information contained in the protein sequence and structure to predict interface residues. Two main approaches are commonly followed to address the interaction sites prediction: structure based (historicallyolder) and sequencedbased depending on which information is at the basis of the analysis at hand. The methods substantially differ because of the underlying training/testing dataset and also because they adopt different definitions of protein surface and contacts, namely residues belonging to the proteinprotein interface. The characteristics of the results of each computational method is strongly dependent on the dataset and on the contact definition adopted [35]. Based on the notion that the data bases of sequences and structures recently underwent an exponential increase, in the following we will detail those methods that more recently can be listed when dealing with the state of the art of prediction of proteinprotein interaction sites. By this we aim at a thorough analysis of all the method recently developed hoping to highlight pitfalls and possible developments to improve the efficacy of the approach. A further subdistinction is made between methods that adopt or do not adopt evolutionary information [35]. Evolutionary based methods are those methods that essentially make use of sequence profile derived from Multiple Sequence Alignments (MSA). All thestateoftheart methods are summed up with their main characteristics in Table 1. 3.1 Sequence Based Methods ISIS (Interaction Sites Identified from Sequence [12]) is a combination of NNs that aims at predicting interacting residues from sequence. The NN input consisted of protein residue composition, sequence profile, predicted secondary structure and solvent accessibility. ISIS first builds the sequence profile required as input by PROF (a predictor of secondary structure from the same group [13]), and secondly the profile and the PROF outputs are given as input to a second NN that classifies each residue as interacting or noninteracting. A residue is defined as belonging to an interface if the distance between any of its atoms is within 6 Å from any atom of a residue in the partner chain. ISIS was trained on a set of 1134 protein chains selected from the PDB and it reached an overall accuracy of 68% (Table 2). 3.2 Structure and Evolutionary Based Methods Recently, the Relative Solvent Accessibility (RSA) of protein residues in a chain has been integrated with high resolution structural data [14] for predicting interacting surface residues. The SPPIDER classifier (Solvent accessibilitybased ProteinProtein Interaction sites IDEntification and Recognition) is a 10 NNs consensus method trained with a cross validated procedure on a 435 protein chains dataset (262 from heterocomplexes and 173 from homocomplexes both derived from PDB). 19 physicochemical features, such as the evolutionary conservation of amino acid properties (type, charge, hydrophobicity) derived from a MSA, were included in the predictor. Nevertheless the authors demonstrated that the most important feature was the difference between the predicted RSA and the real RSA (calculated in the unbound structure), averaged over a window length of 11 residues. The unbound structure was obtained from the complex considering only the chains of interest. The
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Table 1. Recent machine learning methods for proteinprotein interaction sites predictions
Authors (Method)
Surface/ Interface definition
Dataset
Machine learning
Features
Protein: 435 protein AllAtom chains from PDB 19 physicoSurface: (262 from heterochemical RSA > 5% complexes, 173 features: Interface: surface Consensus the most from homoPorollo et al.,2007 residue whose method complexes). important is (SPPIDER) change in RSA is > based on 10 Control set of 149 the WNA over 4% and whose NNs chains (92 from 11 spatial change in exposed heterocomplexes, neighbours of surface upon dSA 57 from homocomplex formation complexes) is > 5 Å Protein: AllAtom Surface: Sequence ASA of at least 1139 nonprofile, ASA, one atom is > 2 Å redundant protein binary profile Dong Q. et al., 2007 SVM Interface: chains from the propensities on residue whose ASA PDB a 12 residues is decreased window > 1 Å upon complexation. Protein: AllAtom Surface: RSA > 15% Chung J. et al., 2006 Interface: residue whose ASA is decreased > 1 Å upon complexation
Li J.J. et al., 2007
Wang B. et al., 2006
Protein: CA trace Surface: RSA > 16% Interface: CACA distance < 1.2 nm Protein: CA trace Surface: RSA > 16% Interface: CACA distance < 1.2 nm
274 nonredundant chains of heterocomplexes taken from the PDB
69 nonredundant chains from ref.10
69 nonredundant chains from ref.10
SVM
Sequence profile, ASA, conservation score on a 12 residues window
LDF
Conservation score of 11 spatial neighbors
SVM
Sequence profile, conservation score of 11 spatial neighbors
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L. Bartoli et al. Table 1. (Continued) 632 nonProtein: redundant protein AllAtom interfaces from Interface: the PDB any pairs of Bordner&Abagyan, (518 homo2005 atoms in each mole dimers, 114 hetcule that are sepaerodimers) rated Testing set: by < 4 Å 43 interfaces from ref .9
Li M. et al., 2007
Chen & Zhou, 2005 (ConsPPISP)
SVM
Local surface properties combined with evolutionary conservation score
Protein: AllAtom Surface: RSA > 15% Interface: Allatom distance < 5 Å
883 chains from heterocomplexes from the PDB
CRF
Sequence profile, ASA, residue conservation and transition features of 15 spatial neighbors
Protein: AllAtom Surface: RSA > 10% Interface: Allatom distance < 5 Å
1156 nonhomologous protein chains (756 homodimers, 400 heterodimers)
Consensus between 6 NNs
Sequence profile, solvent accessibility
Amino acid composition, sequence profile, 1134 protein Ofran & Rost, 2007 predicted NN Protein sequence chains from the (ISIS) secondary, PDB structure solvent accessibility of 9 sequentially neighbors See text for explanations. NN: Neural Network; SVM: Support Vector Machine; CRF: Conditional Random fields; LDF: Linear Discriminant Function ASA: absolute solvent accessibility; WNA: weighted neighbour averages; dSA: difference in solvent accessibility.
observed RSA was calculated with the DSSP program, while the predicted RSA was the result of a previously developed method called SABLE [14]. An interacting residue was defined as a surface exposed residue (RSA > 5%) whose change in RSA between the isolated chain and the complex structure was greater than 4% RSA and whose change in exposed surface area was >5 Å. A filtering procedure for removing negative examples difficult to predict was adopted. An initially “noninteracting” residue was excluded from training if at least 5 of its 10 nearest neighbours belong to the positive class. When tested with a 10fold cross validation the classifier reached an overall classification accuracy of about 74% with a correlation coefficient of 42%.
Prediction of ProteinProtein Interacting Sites
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The sensitivity and specificity were 60% and 64% respectively. This scoring indexes are so far the highest reported (Table 2). Residue interface propensities of different types of complexes (homopermanent, homotransient, heteropermanent, heterotransient) have been analyzed and combined with sequence profiles and accessible surface area (ASA) to recognize binding sites [15] (Dong et al., in Table 1). Sequence profile, ASA and binary profile propensities have been given as input to a SVM system. The training was performed over a nonredundant set comprising 1139 protein chains, chosen from PDB. The binary profile is a 20 valued vector whose elements are 1 or 0, depending on whether the amino acid frequency in a given position of the sequence profile is greater than a given threshold. A residue was defined as belonging to the surface when the ASA of at least one of its atoms was > 2 Å. An interface residue is a surface residue that decreased its ASA more than 1 Å upon complex formation. By this, the best results were obtained when the binary profile propensities were calculated for each class, as detailed above, and assigned to the corresponding class (overall accuracy is 73%, correlation coefficient is 37%); however when the propensities were assigned to a different class (for example heterotransient propensities to heteropermanent complexes) the performance dropped to an overall accuracy of 67% with a 22% correlation coefficient. The specificity and sensitivity were then respectively 40% and 56% (Table 2). Table 2. Scoring indexes of recent machine learning methods for proteinprotein interaction sites predictions
Authors/Method SPPIDER Dong Q. et al., 2007 Chung J. et al., 2006
Li J.J. et al., 2007 Wang B. et al., 2006 Bordner&Abagyan, 2005
Li M. et al., 2007 ConsPPISP ISIS
Accuracy
Sensitivity
Specificity
Correlation coefficient 42% 22% 28% 30%
74% 67% 65%
60% 56% 67% 66% 66%
64% 40% 50% 49% 50%
67%
67%
22%

71% 61% 68%
38% 

30% 
See Table 1 and text for explanations. Scoring indexes are defined in section 4.2.
Chung et al. [16] combined the sequence profile, the ASA and a conservation score for training an SVMbased classifier to identify proteinprotein binding sites. The residue conservation score was the result of a multiple structural alignment, built with the CEMC algorithm, weighted on the crystallographic B factor. This procedure takes into account the structural flexibility than can affect the quality of the alignment. 274 nonredundant chains of heterocomplexes collected from the PDB were included in the dataset. Only surface residues, defined as residues whose RSA was >15%, were considered with a complement of 12 structural surface neighbours. Two residues participate into an interfacial contact if the distance between any of their heavy atoms is less than 5 Å. A predicted noninterface residue becomes an interface residue when the distance
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between its CB atom and the CB atoms of at least three predicted interface residues was < 6 Å. The method achieves a 50% specificity with a 67% sensitivity. Li J.J. et al. [17] represented the evolutionary information through a conservation score based on a phylogenetic tree calculated from the MSA profile. The Rate4Site algorithm was used for the conservation score calculation. By introducing this score, the feature vector is 20 times reduced with respect to a sequence profile based vector. The method was tested on a 69 nonredundant chains obtained by filtering an already compiled dataset [10]. Surface residues were defined as those residues whose RSA calculated with the DSSP was greater than 16%. The protein was represented with its CA trace, choosing a 1.2 nm cutoff for the interaction distance. A Linear Discriminant Function (LDF) was tested on the dataset and reached about 49% specificity, 66% sensitivity (for the positive class) with a correlation coefficient of 28%. In a second paper from the same group, the conservation score was combined with a sequence profile as input for an SVM system [18]. The method with the same input parameters as before reaches a sensitivity of 66.3%, a specificity of 49.7%, an overall accuracy of about 65% and a correlation coefficient of 30%. Bordner and Abagyan [19] described a SVM method trained on a combination of local surface properties and an evolutionary conservation score. A newly developed Bayesian algorithm that made use of phylogenetic trees was adopted to calculate the evolutionary score. Local surface patches of 15 residues (central residue and 14 surface neighbors obtained comparing CA distances) were generated for each surface residue. Zscores of residue frequencies in multiple alignment and evolutionary rates for each residue of the patch were adopted as SVM input. The system was trained on a 1494 nonredundant protein interfaces dataset (518 homodimers, 114 heterodimers and 862 multimers from the PDB). A post processing procedure was then applied to remove possible false positives: residues were clustered in patches comprising those whose CA atoms were less than 6 Å apart; predicted interface patches were changed in noninterface ones if their ASA was 90% of the sequences. Choosing a representative for each cluster the final dataset comprised a nonredundant set of heterocomplexes with 95% over 90% of the sequences length, the complex was identified as an homocomplex and discarded. For chains interacting with multiple partners, only the one with the most partners was chosen as representative. A further alignment with the CE_MC algorithm choosing the homologous at the SCOP family level, retained only the chains that were aligned over 60% of their length and with a Zscore >4.0. The final dataset contained 274 nonredundant chains of heterocomplexes. The Li J.J. et al. [17] dataset comprises only 69 chains with sequence identity less than 30%. Starting from the PDB Bordner and Abagyan [19] selected all Xray structures and used the biological unit information of the mmCIF file to build the structure of the complex. Two chains in a complex were defined as interacting if heavy atoms in each chain were closer than 4 Å with heavy atoms on the other chain. Pairs containing chains shorter than 20 residues were removed. The quality of the dataset was improved checking the consistency between PDB file and the SwissProt annotation. In order to remove homocomplexes sequences were clustered and only those with less than 30% sequence identity were retained (BLAST search against nr with Evalue cutoff 0.1). A further filter was applied for removing interfaces showing, according to PDB file information, mutations and immune system proteins with polymorphisms and somatic mutations. Only interfaces larger than 10 residues were retained in the last 1494 proteinprotein interfaces dataset, composed by 518 homodimers, 114 heterodimers and 862 multimers. Different interfaces for the same chain are considered in an independent way. Li M. et al. [20] considered the July 2005 PDB release. The resolution cutoff value chosen was 3.5 Å. Redundancy was avoided by retaining only those structures whose sequence identity was ≥ 30% over > 90% of the sequence, ending up with 1276 chains from heterocomplexes. In general, complexes can have more than one chain that can form more than one interface. For addressing this problem each chain was considered once and only one partner with the most interfacial residues was considered but also other cases were taken into account. Indeed minor interfaces were labeled as interface, noninterface or even excluded from the dataset and for each case a different result was reported. Chen and Zhou [22] substantially implemented the same method of 2001 with a bigger database extracted from the January 2002 PDB release. All multiple chain structures, whose chains were longer than 40 residues and whose interface was larger than 20 residues were considered. Each chain was assigned only to the partner with the largest interface. Residues belonging to other interfaces were labeled as non interface residues or eliminated from the training set. The similarity cutoff for the PSIBLAST allagainstall search was 30% over the aligned region that had to cover at least 90% of the two sequences. A homodimer was defined as a complex with highly homologous chains (95%) over a 90% aligned region. Their set comprises 798 homodimers and 458 heterodimers.
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5 Our Approach: ISPRED In the following we introduce the method that we implemented for the prediction of proteinprotein interaction sites. The method although, described before [10], is now retrained on new training set. The method is essentially a feed forward neural network trained with a standard back propagation algorithm. Each protein residue is represented with its carbon alpha (CA). By this, the protein surface is represented using its CA trace and the interface residues are defined as those surface residues (residues with relative solvent accessibility > 16%) whose Euclidean distance with at least one residue of the partner chain is below 1.2 nm. Solvent exposure is separately computed for each chain, using the DSSP program. This threshold value is selected after comparison with the patches obtained using an allatom representation of the residues. The method exploits the evolutionary information contained in the multiple sequence alignment of protein families, encoded in the sequence profile. The profile is a L*20 matrix, where L is the protein sequence length, that for each position of the alignment contains the frequency of occurrence of each type of residue in the corresponding position of the alignment. The system is trained using an 11 residuelong window centered on the surface residue to be predicted with 10 nearest neighbors in the surface patch. To give a rough approximation of the local surface, the residues included in the input window are close in space and not necessarily adjacent in the sequence and represent. Thus, each residue in the input window is encoded as a vector of 20 elements, whose values are the frequencies extracted fro the sequence profile. Each protein residue is thus encoded in a 232 nodes input vector that contains the sequence profile of the residue to predict, the sequence profile of its 10 nearest spatial neighbors, the relative accessible surface area of the residue to predict and the frequency of interaction of the residues in the window (namely 11), calculated on the overall dataset. The output of the neural network is the probability of each surface residue to be in an interaction site given the current input. Neural network predictions are uncorrelated. In order to take into account the cooperation among close surface residues, we averaged the network output over the list of neighbors. The probability for a given residue is calculated as follows: P(i) = Σj w(i,j) O(Ri(j))/ Σj w(i,j) where O(Ri(j)) is the output corresponding to the neighbor j of the residue i and w(i,j) is a weight. We tested several weighting schemes. The uniform distribution (Uniform) wU(i,j)=1 the exponential decreasing with the Euclidean distance between the residue and its neighbors (Exp), wE(i,j)=exp[d(i, Ri(j))] the decreasing with the inverse of the Euclidean distance,between the residue and its neighbors (Inv),
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wI(i,j)=I/[ d(i, Ri(j))(iδ(0,j)+ δ(0,j))] With the inclusion of Kronecker delta (δ), w(i,j)=1 when we refer to the current residue (j=0) otherwise it decreases with the inverse of the distance. We selected the uniform distribution as the best performing one. The introduction of this smoothing function reduces the number of spurious assignments and for this reason it improves the predictor performance [11]. 5.1 ISPRED2.0: Dataset Description Our dataset is composed of 626 protein chains belonging to 318 different protein complexes (291 dimers and 27 multimers) obtained from the PDB. Among the 116 homocomplexes, 40 structures are transient and 76 are obligate, while among the 202 heterocomplexes, 106 are classified as transient and 106 as obligate. For the selection of the transient complexes we found that 215 of the selected chains have a correspondent structure resolved both in complex and in a complex independent stable structure. Only the complexes with at least one monomer with a highly homologous (≥70%) unbound form have been selected as transient complexes. We also filtered out structures whose sequences are shorter than 40 residues. For purpose of comparison with SPPIDER, we randomly selected a benchmark subset of 50 protein chains from our initial dataset, with the constraint that they do not share more than 25% sequence similarity with any of the sequences in the Porollo dataset [14]. 18 chains belong to homocomplexes (2 transient complexes and 16 obligate) and 32 are from heterocomplexes (among with 16 transient). We also selected a subset of 417 proteins (S417) that are the complement of the intersection (S467) between the 626 proteins (S626) dataset and the complete Porollo’s dataset, filtered out from the 50 proteins selected as described above. The intersection between the two sets has been computed with a BLAST alignment. 5.2 Scoring the Performance The results of the methods are evaluated using the following measures. The overall accuracy (Q2) is defined as: Q2= Overall accuracy = TP+TN/(TP+TN+FP+FN) where:  TP = number of true positive predictions (observed interacting residues predicted as interacting)  TN = number of true negative predictions (observed noninteracting residues predicted as noninteracting)  FP = number of false positive predictions (observed noninteracting residues predicted as interacting)  FN = number of false negative predictions (observed interacting residues predicted as noninteracting) The sensitivity (coverage) for the positive class Q[+] and for the negative one Q[] are computed as follows Q[+]= TP/(TP+FN) Q[] = TN/(TN+FP)
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To measure the probability of correct predictions, we computed the specificity (namely the accuracy for each class, + and ) expressed as follows: P[+] = TP/(TP+FP) P[] = TN/(TN+FN) The correlation coefficient is: C = (TP*TNFP*FN)((TP+FP)*(TP+FN)*(TN+FN)*(TN+FP))1/2
6 Results The benchmark of the new release of ISPRED (2.0) is done considering the predictor with the highest score around: SPPIDER. The problem that we want to tackle is how good is SPIDDER performance when tested on different sets from the one used by the authors. In the first experiments ISPRED 2.0 is directly compared with the performance of SPPIDER, as described in literature. Results are shown in Table 3, where also a second cascaded network is added to our predictor to filter out non correct assignments. Results are obtained by averaging over 10 different values for each scoring index computed by means of a cross validation procedure. SPIDDER was originally trained on 435 protein complexes (S435) and tested on 149 proteins (S149). In the last experiment of Table 3 ISPRED was tested on the SPPIDER testing set, again adopting a cross validation procedure. It is evident that a second cascaded network slightly improves our performance. Values however are still lower than that declared for SPPIDER. As discussed above (section 4) we are now comparing results obtained with different training and testing methods. We then used datasets comprising 50 randomly selected proteins from the 626 of our data set and we tested SPPIDER performance. The results are reported in Table 4. It evident that on small testing sets, different from the original one, SPPIDER performances are well below the reported values. This is so even when consider independent test sets of larger dimension never seen before by the two methods (Table 5). All the PDB proteins were filtered with ISPRED to predict putative contact sites. The predictions are available at our web site with the scores obtained for each putative contact residue. As an example, Figure 1 shows a protein 3D structure with highlighted the predicted interacting sites localized on surfaces. Table 3. ISPRED 2.0 at work
Method SPPIDER* ISPRED2.0 ISPRED2.0^ +2nd Net ISPRED2.0# +2nd Net
Q2 0.74 0.66 0.68
Q[+] 0.60 0.54 0.58
P[+] 0.64 0.7 0.72
Q[] 0.77 0.77
P[] 0.62 0.65
C 0.42 0.32 0.36
0.67
0.58
0.76
0.71
0.65
0.35
* * from [14]. ^with a second cascaded neural network. #trained on the 435 and tested on S149. ISPRED 2.0 is trained on S626 and tested on S149.
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L. Bartoli et al. Table 4. SPPIDER tested on randomly selected protein sets
Dataset* #1 #2 #3 #4 #5 AVERAGE
Q2 0.69 0.67 0.69 0.70 0.66 0.68
Q[+] 0.50 0.50 0.51 0.53 0.53 0.51
P[+] 0.43 0.44 0.43 0.46 0.48 0.45
Q[] 0.76 0.74 0.76 0.77 0.72 0.75
P[] 0.81 0.79 0.81 0.82 0.76 0.80
C 0.25 0.24 0.26 0.29 0.24 0.26
*see Data set description Table 5. Comparison of SPPIDER and ISPRED2.0 performance
Method SPPIDER* ISPRED2.0° ISPRED2.0+ ISPRED2.0# +2nd Net
Q2 0.68 0.66 0.66 0.67
Q[+] 0.54 0.45 0.49 0.53
P[+] 0.47 0.43 0.44 0.45
Q[] 0.74 0.74 0.73 0.73
P[] 0.79 0.78 0.77 0.79
C 0.28 0.21 0.22 0.25
*tested on S417. °trained on S626, tested on S417. +trained on S435, tested on S467. #trained on S435, tested on 467.
Fig. 1. Example of prediction of the protein 1ibc (inhibited interleukin—1beta conerting enzyme), obtained with ISPRED 2.0 and stored in the ZenPatches database (http:// gpcr.biocomp.unibo.it/biodec/). In red are the true positives, in green false negatives and in yellow false negative predictions.
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7 Conclusions As a final consideration we can observe the recent methods focusing on predictions of interacting sites in proteins have exploited a large collections of chemicophysical properties for generating the input code. By this different rules have been described and obtained from different data bases. This obviously hampers a direct comparison of the different methods including ours. Even though standard rules are generally accepted when implementing machine learning approaches, different inputs apparently end up in different scoring indexes. In this paper we focused on the specific problem of the data bases. We addressed the question of how two highly performing predictors can be compared when they have been trained with different criteria and data bases. We observe that when data bases are randomly selected the performance is rather similar suggesting that possible improvements described in literatures [14] may be due to overfitting on a specific data base. It is therefore necessary to standardize both training and testing sets for implementing predictors that later on may be benchmarked on more rigorous criteria. In any case the actual state of the art predictors seem to have reached an upper limit scoring performance that at least in our case on a more abundant data set of examples. A previous implementation of ISPRED [10], trained and tested on a less abundant data set scored with a 26% correlation index. This value, although obtained in 202, is similar to the most recent ones shown in Table 2. Our experiments indicate that an upper bound to the scoring indexes is possibly the one declared by SPPIDER, that we have shown is wuite sensible to different data sets, as opposite to ISPRED 2.0. Also the method seems not be an issue as recently demonstrated with a SVM based method that scored as high as SPPIDER (with a coefficient correlation of 33%) on a similar data set [5]. The problem of how to treat multiple interfaces then emerges. According to a molecular view of proteinprotein interaction it should be considered that one protein may be even involved in different protein complexes, although in a transient way. Then the question poses as to whether we really are in the position of correctly describing an interacting surface, when it is possible that presently the PDB collects structural information only of those complexes whose life time is compatible with the crystallization procedures. When the same or a similar protein are interacting with different partners, are these patches to be included or excluded? According to a molecular view of proteinprotein interaction it should be considered that one protein may be even involved in different protein complexes, although in a transient way. Then the question poses as to whether we really are in the position of correctly discriminating false positives. It may be possible that considering protein complexes solved with atomic resolution we are missing other important features of protein surfaces that may help us in improving the gap among protein interacting networks and molecular recognition at the protein level. For the time being we can rely on putative interaction sites as those collected in the ZenPatches database (http:// gpcr.biocomp.unibo.it/biodec/) that can help in supporting experimental large scale data and at the molecular level in designing further experiments for validation studies.
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Acknowledgements RC acknowledges the receipt of the following grants: FIRB 2003 LIBIInternational Laboratory of Bioinformatics and the supportto the Bologna node of the Biosapiens Network of Excellence project within the European Union’s VI Framework Programme (contract number LSGHCT2003503265).
References 1. Chothia, C., Janin, J.: Principles of proteinprotein recognition. Nature 256, 705–708 (1975) 2. Jones, S., Thornton, J.M.: Prediction of proteinprotein interaction sites using patch analysis. J. Mol. Biol. 272, 133–143 (1997) 3. Zhou, H., Qin, S.: Interaction site prediction for protein complexes: a critical assessment. Bioinformatics 21, 2469–2501 (2007) 4. De Vries, S.J., Bonvin, A.M.: How proteins get in touch: interface prediction in the study of biomolecular complexes. Curr. Protein Pept. Sci. 9, 394–406 (2008) 5. Ezkurdia, I., Bartoli, L., Fariselli, P., Casadio, R., Valencia, A., Tress, M.L.: Progress and challenges in predicting proteinprotein interaction sites. Briefings in Bioinformatics 10(3), 233–246 (2009) 6. Fu, H.: ProteinProtein Interactions: Methods and Applications (Methods in Molecular Biology). Humana Press Inc., New York 7. KatchalskiKatzir, E., Shariv, I., Eisenstein, M., Friesem, A.A., Aflalo, C., Vakser, I.A.: Molecular surface recognition: determination of geometric fit between proteins and their ligands by correlation techniques. Proc. Natl. Acad. Sci. USA 89, 2195–2199 (1992) 8. Berezin, C., Glaser, F., Rosenberg, J., Paz, I., Pupko, T., Fariselli, P., Casadio, R., BenTal, N.: ConSeq: the identification of functionally and structurally important residues in protein sequence. Bioinformatics 20, 1322–1324 (2004) 9. Nooren, I., Thornton, J.M.: Structural characterization and functional significance of transient proteinprotein interactions. J. Mol. Biol. 325, 991–1018 (2003) 10. Fariselli, P., Pazos, F., Valencia, A., Casadio, R.: Prediction of proteinprotein interaction sites in heterocomplexes with neural networks. Eur. J. Biochem. 269, 1356–1361 (2002) 11. Fariselli, P., Zauli, A., Rossi, I., Finelli, M., Martelli, P.L., Casadio, R.: A neural network method to improve prediction of proteinprotein interaction sites in heterocomplexes. In: IEEE International Workshop on Neural Network on Signal Processing 2003, Toulouse, France, pp. 33–41. IEEE Press, Los Alamitos (2003) 12. Ofran, Y., Rost, B.: ISIS: interaction sites identified from sequence. Bioinformatics 23, 13–16 (2007) 13. Rost, B.: Prediction in 1D: Secondary structure, membrane helices, and accessibility. In: Bourne, P., Weissig, H. (eds.) Structural bioinformatics. WileyLiss, Hoboken (2002) 14. Porollo, A., Meller, J.: Predictionbased fingerprints of proteinprotein interactions. Proteins 66, 630–645 (2007) 15. Dong, Q., Wang, X., Lin, L., Guan, Y.: Exploiting residuelevel and profilelevel interface propensities for usage in binding sites prediction of proteins. BMC Bioinform. 8, 147 (2007) 16. Chung, J., Wang, W., Bourne, P.E.: Exploiting sequence and structure homologs to identify proteinprotein binding sites. Proteins 62, 630–640 (2006)
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17. Li, J.J., Huang, D., Wang, B., Chen, P.: Identifying proteinprotein interfacial residues in heterocomplexes using residue conservation scores. Int. J. Biol. Macromol. 38, 241–247 (2006) 18. Wang, B., Chen, P., Huang, D., Li, J.J., Lok, T.M., Lyu, M.R.: Predicting protein interaction sites from residue spatial sequence profile and evolution rate. FEBS Lett. 580, 380– 384 (2006) 19. Bordner, A.J., Abagyan, R.: Statistical analysis and prediction of proteinprotein interfaces. Proteins 60, 353–366 (2005) 20. Li, M., Lin, L., Wang, X.L., Liu, T.: Proteinprotein interaction site prediction based on conditional random fields. Bioinformatics 23, 597–604 (2007) 21. Zhou, H.X., Shan, Y.: Prediction of protein interaction sites from sequence profile and residue neighbor list. Proteins 44, 336–343 (2001) 22. Chen, H., Zhou, H.: Prediction of interface residues in proteinprotein complexes by a consensus neural network method: test against NMR data. Proteins 61, 21–35 (2005)
The Equivalence between Biology and Computation John K. Heath School of Biosciences, University of Birmingham, UK
[email protected] Abstract. A major challenge in computational systems biology is the articulation of a biological process in a form which can be understood by the biologist yet is amenable to computational execution. Process calculi have proved to especially powerful computational tools for modelling and reasoning about biological processes and we have previously described, and implemented, a Narrative approach to describing biological models which is a biologically intuitive high level language that can be translated into executable process calculus programs. Here we discuss an extension to the narrative approach which attempts to directly link biological data with Narrative primitives by suggesting an equivalence relationship between a string (the amino acid sequence) and a process. We outline future challenges in applying this approach more generally.
1
Introduction and Motivations
There are multiple incentives for a convergence between computing and biology [1]. Classically this has been the adoption by biologists of computational tools to store, search, visualise and interrogate large sets of biological data. However the relationship can be taken further by the aspiration to realise a biological process in the form of a computer program. The predicted beneﬁts of this challenge are many. A pragmatic motivation for the biologist is that it could accelerate the pace of biological discovery, help design better laboratory experiments and reason more clearly about biological processes [2]. But there are greater wins to be had. A program that faithfully emulated biology could be used to study processes which cannot practically be investigated in the biologists laboratory. Examples would include biological processes that take place over very long periods of time such as pathway evolution, aging and degeneration, or the accumulation of mutations in the lifetime of a tumour. Programs could be used to study the outcomes of experiments that can be conceived but cannot be executed in the laboratory. Programs could be used to even identify experiments that biologist cannot conceive. Computer programs that behaved like biological processes could revolutionise the development of new therapies by allowing exhaustive exploration of treatment scenarios without recourse to laboratory investigation and will be essential to realise the goal of personalised medicine: the program can be employed to predict the outcome of therapy. P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 18–25, 2009. c SpringerVerlag Berlin Heidelberg 2009
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Finally programs that behaved like biology could be employed to design new biological processes with speciﬁed properties including, for example, biology that behaved like a computer. Realisation of this challenge has outcomes for computer science in that it demands programs that do not simply emulate the system but in fact are the system. This calls for the development of computational techniques which exactly and formally behave like a biological process. This represents true convergence of the two disciplines. This paper reviews current progress towards this aim with a speciﬁc emphasis on the equivalence between biological data types and events and computational formalisms.
2
Algorthimic Systems Biology
The classical, and most widely employed, approach to modelling biological processes has been via the use of mathematical tools that express the process in the form of numerical values which are related to each other in the form of equations. The behaviour of the process can then be simulated by calculating the solution to the equations which plots the changing relationship between the numerical variables over time. In this fomalism the computer is a calculating device [3,4]. In the past few years a diﬀerent use of computers has been proposed. This springs from the fundamental properties of computation in which the state of the machine changes dynamically according to a set of rules (algorithims). Thus in computation the system passes through state changes according to the instructions supplied. In its most general sense the algorthimic approach to biology deﬁnes the states of a biological system and the instructions that relate the states to each other. Thus the central questions in this light are how is a biological state deﬁned and what are the relationships between these states in the biological system. Thus in order to program the computing device to behave like a biological process we require a method which maps biological properties onto computational primitives. Such a mapping has to be rigorously speciﬁed and therefore requires some form of operational semantics which exists in the realm of biological data types. 2.1
Process Calculi
One of the most popular manifestations of the algorithmic approach to biology has been the use of process calculi which are set of approaches developed to study concurrent communicating systems [5]. The application of process calculi to biological systems was ﬁrst advocated by Regev et al [6] who drew attention to the formal similarity between core concepts in process calculus languages and biological processes. These are: the representation of biological components as processes; the interaction between biological entities as a communication between processes and that the consequence of a communication event results in a change of state (ie the spawning of a new process). Process calculi provide a rich
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and expressive operational semantics for describing concurrent communicating systems with a small number of operational primitives and rigorous logical rules for manipulating the primitives allowing the computer scientist to reason about the processes embodied in the program. A number of diﬀerent process calculus approaches have now been applied to the study of biological systems [3]. Of key importance it has been possible to demonstrate for a number of biological processes that the process calculus approach yields programs that exhibit realistic biological behaviours which can be successfully employed to reason about biological system and successfully predict the outcome of real (or hypothetical) biological interventions [2]. These outcomes are suﬃciently encouraging to consider the further convergence between process calculus concepts and their biological counterparts. 2.2
Biological Narratives
In considering how to pursue the convergence of computer science and biology we need to consider the current process by which a biological scheme is executed in the form of a program. We shall here conﬁne our analysis to the study of signal transduction pathways which have so far been the major class of biological problem investigated by process calculus tools. Biologists traditionally express their hypotheses in the form of informal diagrams. These are informal in the following sense: they employ a variety of notational styles which can be ambiguous or confusing to the non biologist (although various formal notations have been proposed there is no current universal standard [7]). The diagrams are abstractions in that they are generally designed to emphasise some particular feature of the pathway and components that are not relevant to the immediate problem are often, for the sake of clarity, eliminated. It is therefore frequently confusing for the non biologist to ﬁnd seemingly the same process articulated by quite distinct diagrams. Finally diagrams are inherently static representations and dynamic features of pathway behaviour are diﬃcult to represent. For these reasons the computer scientist can ﬁnd the translation of a biological process into a computational formalism confusing and ambiguous. The more productive approach to date has been true collaboration between the computer scientist and biologist in which a dialogue results in the formulation of the computer program in an appropriate language for execution. The dialogue is required because few (if any) biologists have the training or motivation to formulate their ideas in the form of speciﬁc computationalorientated semantics. The risk here is that the process of translation results in loss of information for successful articulation of the biological pathway. We [8,9] have developed an approach which aims to address these issues by developing a semantic system which is aimed to be intuitive to the biologist, suﬃciently expressive to faithfully articulate the biological process and yet readily translatable into a process calculus language. The basic components of the language are molecules (components) which are located in speciﬁed locations (compartments). The components can interact (for example bind or unbind) undergo modiﬁcations (eg phosphorylation/ dephosphorylation) and move between
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compartments. As a result of interactions components can change in number (ie synthesis or degradation). These basic elements (which map directly onto the underlying process calculus primitives) are then formulated as a Narrative of events. The Narrative thus represents the instructions that specify the biological process under investigation. In accord with the process calculus approach steps in the Narrative can be concurrent and exhibit dependencies: for example a binding event may be dependant on a prior phosphorylation event or relocation between compartments may require speciﬁc complexes to be formed. A number of pathways have been formulated in a Narrative form [2,9] and been shown not only to produce outputs which resemble real world biological data, but, by removal of components in the narrative or alteration in rate parameters, permit the biologist to examine the consequences of perturbations on the behaviour of the model. The advantage of a high level, biologically intuitive and formally rigorous language is that models can be speciﬁed by the biologist without them needing to understand the details of the underlying computational execution. The availability of translators from the Narrative language to diﬀerent process calculus languages should also allow the same model to be executed by diﬀerent process calculus languages.
3
Equivalence between Biology and Computation
The Narrative approach illustrates a method of articulating a biological process in a way which is formally rigorous for computation but intuitive to the biologist. However there is still a gap between the realm of biological data and the computational model as the approach still requires the biologist to interpret their laboratory data in the form of a Narrative description. We now suggest an extension to the narrative approach in which the basic elements of the model are direct representations of laboratory data leading to a formal equivalence between biological data and computational execution. For this purpose we need to understand the type of data generated in a typical biology experiment. Again by way of illustration we shall conﬁne our discussion to the analysis of signal transduction pathways. 3.1
Components
In the computational model the component is simply a name with speciﬁed properties. However the name points to a real biological molecule with experimentally deﬁned attributes. The corresponding component in the biological domain is a string: the amino acid sequence of the protein. Replacing the component with the string therefore directly connects the biological entity with the computational primitive. This opens up two avenues: the ability to specify the component in the model with veriﬁed biological data and a direct connection between model primitives and experimental biological data. Thus for example in the model of Guerrerio et al [9] gp130 refers to a real biological molecule with known and veriﬁable biological attributes. These are stored
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in various biological databases such as Uniprot [10] (http://www.uniprot.org). Thus replacing the name with a database pointer UniProtKB/SwissProt P40189 automatically connects biological knowledge with the computational formalism. The information collected in the biological database contains all the details required for speciﬁcation of the component in the model such as subcellular location (or compartment), sites of post translational modiﬁcation (ie potential state changes), domain architecture and primary sequence. In fact direct exploitation of biological database information to deﬁne a component in a model in most cases will provide much more information than the modeller would require. The amino acid (or nucleotide) sequence present in the database also provides the means for biological veriﬁcation. Thus the existence of a component in the biological domain can be veriﬁed by unique barcodes either in the form of the component itself or by sequence tags added by genetic engineering techniques [11]. The unique biological identiﬁer can also be extended to components that exist in certain states for example speciﬁc phosphorylated forms of the protein can be identiﬁed by phospho speciﬁc antibodies [12]. In addition using for example fusions with intrinsically ﬂuorescent proteins or antibody localisation the location of the component can also be experimentally veriﬁable. There is thus a direct map between the component in the model and the experimental data generated by the biological investigator. By this means it becomes possible to directly connect an observable eg the amount or location of a component and its system state to the model. 3.2
Complexing Rules
The formation and destruction of molecular complexes is central to most signal transduction processes. From the computational perspective these are processes spawned from the merger of two previous processes. In the biological data the existence of complexes can be detected by a variety of experimental techniques. These include coimmunoprecipitation, colocalisation by microscopy or via proteomics techniques such as twohybrid interactions or mass spectroscopy [13]. These data types have also been curated in biological protein/ protein interaction databases databases such as String [14] (http://string.embl.de/). Thus a binding event in the model can be linked to the existence of an experimentally veriﬁed binding event in the database as a concatenation of two (or more) strings. In addition as the model is run computationally it will generate outputs which refer to a potential biological experiment of the form does a complex between component A and component B exist at time T in compartment X? Thus connecting the biological data and the computational primitive allows the model to yield outputs which describe the outcomes of a hypothetical biological experiment. 3.3
Compartments and Space
A notion of conﬁned space was introduced in the narrative approach as an extension to the original speciﬁcations of Regev et al [15] in order to more correctly
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reﬂect biological reality in which the sequestration of biological processes, for example inside membrane conﬁned spaces such as the nucleus or the golgi, plays a key role in the process under investigation [review]. Some biologically orientated process calculus languages such as Bioambients [15] have been speciﬁcally devised with compartments in mind and used to model signalling processes based upon directed traﬃcking of components between membranebound compartments. Indeed we have been able to show in modelling the gp130/Jak/STAT pathway using the Narrativebased process calculus approach that the rate of movement between cellular compartments (in this case nucleus and cytoplasm) is a key determinant of pathway behaviour [9]. We have also been able, using ambients, to model the traﬃcking of cell surface receptors into diﬀerent cytosolic compartments which compares well with biological data [16]. Extensions to this approach in which traﬃcking of receptors is reactive to changes in the states of components in the model (eg dependent upon a phosphorylation event) are under development (M.G. Vigliotti, S.van Bakel, J. K. Heath unpublished). These results are encouraging but need to be further developed if a full equivalence between biological datasets and computational models is to be achieved. In particular to fully articulate the richness of biological data some explicit notation for spatial localisation is required. Experimental biological investigations of signalling pathways increasingly exploit the power of live cell imaging [17]. In this technique proteins are visualised by fusion to an endogenously ﬂuorescent reporter protein such as Green Fluorescent Protein (GFP) and tracked dynamically in individual cells by confocal light microscopy. The biological data in this case is in the form of a video and is analysed a single cell level which is especially appropriate for modelling as each cell is in essence an individual execution of the model rather than averaged over many cells as would be the case for methods such as western blotting. The development of computational techniques for extracting and classifying data from live cell imaging is a rapidly developing area of computer science. The application of this general approach is dramatically transforming our biological understanding of the spatial architecture of cell signalling pathways. Thus the commonly modelled ras/raf/mapk pathway is usually modelled as a connected set of binding and phosphorylation events occurring within a single compartment [18]. The imaging data reveals however that this module is highly dynamic with components of the pathway shuttling between diﬀerent compartments as the signal is propagated [19]. There are also important problems in biology such as the molecular machines controlling cell motility or mitosis  in which molecular movement in time and space is the essence of the problem.
4
Conclusions
Process calculi concepts and approaches are now established as very well suited to modelling biological pathways by the algorithimic approach. An obstacle to more widespread adoption has been the language gap between computational and biological formalisations. High level, biologically intuitive languages as illustrated by the Narrative approach undoubtedly help to bridge this gap. However
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biological investigation is ultimately driven by data and formalisms for directly mapping biological data types into executable programs is the next step in the convergence of biology and computing.
Acknowledgements Research in the authors laboratory is supported by Cancer Research UK and the FP6 Endotrack programme.
References 1. Priami, C.: Algorithmic Systems Biology An Opportunity for Computer Science. Communications of the ACM 52, 80–88 (2009) 2. Kwiatkowska, M.Z., Heath, J.K.: Biological pathways as communicating computer systems. J. Cell Sci. (in press, 2009) 3. Fisher, J., Henzinger, T.A.: Executable cell biology. Nat. Biotechnol. 25(11), 1239– 1249 (2007) 4. Hunt, C.A., Ropella, G.E., Park, S., Engeleberg, J.: Dichotomies between computational and mathematical models. Nature Biotechnology 26, 737–738 (2008) 5. Milner, R., Parrow, J., Walker, D.: A calculus of mobile processes, Pts 1 and 2. Information and computation 100(11), 1–40 (1992) (print) 6. Regev, A., Silverman, W., Shapiro, E.: Representation and simulation of biochemical processes using the picalculus process algebra. In: Paciﬁc Symposium Biocomput., pp. 459–470 (2001) 7. Matsuoka, Y., Ghosh, S., Kitano, H.: Consistent design schematics for biological systems: standardization of representation in biological engineering. J. R Soc. Interface (June 3, 2009) 8. Guerriero, M.L., Heath, J.K., Priami, C.: An automated translation from a narrative language for biological modelling into process algebra. In: Calder, M., Gilmore, S. (eds.) CMSB 2007. LNCS (LNBI), vol. 4695, pp. 136–151. Springer, Heidelberg (2007) 9. Guerriero, M.L., Dudka, A., UnderhillDay, N., Heath, J.K., Priami, C.: Narrativebased computational modelling of the Gp130/JAK/STAT signalling pathway. BMC Syst. Biol. 3, 40 (2009) 10. Jain, E., Bairoch, A., Duvaud, S., Phan, I., Redaschi, N., Suzek, B.E., Martin, M.J., McGarvey, P., Gasteiger, E.: Infrastructure for the life sciences: design and implementation of the UniProt website. BMC Bioinformatics 10, 136 (2009) 11. Puig, O., Caspary, F., Rigaut, G., Rutz, B., Bouveret, E., BragadoNilsson, E., Wilm, M., Sraphin, B.: The tandem aﬃnity puriﬁcation (TAP) method: a general procedure of protein complex puriﬁcation. Methods 24(3), 218–229 (2001) 12. Chung, A.S., Chin, Y.E.: Antibody array platform to monitor protein tyrosine phosphorylation in mammalian cells. Methods Mol. Biol. 527, 247–255 (2009) 13. Albeck, J.G., MacBeath, G., White, F.M., Sorger, P.K., Lauﬀenburger, D.A., Gaudet, S.: Collecting and organizing systematic sets of protein data. Nat. Rev. Mol. Cell Biol. 7(11), 803–812 (2006) 14. Jensen, L.J., Kuhn, M., Stark, M., Chaﬀron, S., Creevey, C., Muller, J., Doerks, T., Julien, P., Roth, A., Simonovic, M., Bork, P., von Mering, C.: STRING 8–a global view on proteins and their functional interactions in 630 organisms. Nucleic Acids Res. 37(Database issue), D412–D416 (2009)
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15. Regev, A., Panina, E.M., Silverman, W., Cardelli, L., Shapiro, E.: Bioambients: An abstraction for biological compartments. Theoretical Computer Science, 141–167 (2004) 16. Heath, J.K., Khan, I., van Bakel, S., Vigliotti, M.G.: Modelling intracellular fate of FGF receptors with BioAmbients. In: Proceedings of Int. Workshop Quantitative Aspects of Programming Languages (QAPL 2008). ENTCS, vol. 220, pp. 181–197 (2008) 17. SabouriGhomi, M., Wu, Y., Hahn, K., Danuser, G.: Visualizing and quantifying adhesive signals. Curr. Opin. Cell Biol. 20(5), 541–550 (2008) 18. Calder, M., Gilmore, S., Hillston, J.: Modelling the inﬂuence of RKIP on the ERK signalling pathway using the stochastic process algebra PEPA. In: Priami, C., Ing´ olfsd´ ottir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on Computational Systems Biology VII. LNCS (LNBI), vol. 4230, pp. 1–23. Springer, Heidelberg (2006) 19. Lu, A., Tebar, F., AlvarezMoya, B., LpezAlcal, C., Calvo, M., Enrich, C., Agell, N., Nakamura, T., Matsuda, M., Bachs, O.: A clathrindependent pathway leads to KRas signaling on late endosomes en route to lysosomes. J. Cell Biol. 184(6), 863–879 (2009)
BlenX4Bio – BlenX for Biologists Corrado Priami1,2 , Paolo Ballarini1 , and Paola Quaglia1,2 1 CoSBi, Italy University of Trento, Italy {priami,ballarini,quaglia}@cosbi.eu 2
Abstract. We introduce BlenX4Bio, a highlevel interface for the programming language BlenX. BlenX4Bio allows biologists to write BlenX programs without having any programming skills. The main elements of a biological model are speciﬁed by ﬁlling in a number of tables. Such tables include descriptions of both static and dynamic aspects of the biological system at hand and can then be automatically mapped to BlenX programs for simulation and analysis by means of the CoSBi Lab software platform. In this paper we illustrate the main characteristics of BlenX4Bio through examples taken from biology textbooks.
1
Introduction
It is well accepted that modern biology, and mainly systems biology, has to have modeling techniques in its toolset. The ever increasing complexity of the modelled systems pose a challenge to biological modelling and the classical equationbased (both mathematical and chemical) modeling approaches, in particular, proved to suﬀer from the combinatorial explosion of the size of the model. The recent ﬁeld of algorithmic systems biology [12] proposes computer science as a foundation for systems biology and as a means which, relying on basic principles of programming language theory, allows for the speciﬁcation of complex systems in a concise and precise manner. Algorithms force modelers and biologists to think about the mechanisms that govern the behavior of the system they are studying. Algorithms are highlevel expressions of a system’s behaviour which describe the states in which a system can be and the conditions under which a system evolves from one state to another. Computer science relies on programming languages to unambiguously describe and execute algorithms. Furthermore, concurrency must be a core design principle of any foundational formalism for modern biology, given the importance of simultaneous interactions in biological phenomena. Concurrent programming languages can easily and eﬃciently express the mechanistic rules that propel algorithmic systems biology as they already proved to be eﬀective in modeling, analyzing and programming large computer networks. Algorithmic systems biology must equip concurrent programming with an highlevel interface simpliﬁed to the extent that biologists can use it without knowing the basics of programming. As a consequence we designed BlenX4Bio so to hide all programming based details from its users. BlenX4Bio is strongly P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 26–51, 2009. c SpringerVerlag Berlin Heidelberg 2009
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inspired by molecular cell biology and by the narrative language which was introduced in [6,4] as a biologistsfriendly modeling formalism. BlenX4Bio is targeted and optimized for the BlenX language [3] from which it inherits the modeling philosophy. The use of programming languages to model biological systems is an emerging ﬁeld that enhances current modeling capabilities (richness of aspects that can be described as well as easiness, composability and reusability of models) [13]. The metaphor which inspires this idea is one where biological entities are represented as programs being executed simultaneously and the interaction of two entities is represented by the exchange of a message between the programs representing those entities in the model [15]. The biological entities involved in the biological process and the corresponding programs in the abstract model are in a onetoone correspondence, thus coping by construction with the combinatorial explosion of variables needed in the equational approach to describe the whole set of states through which a single component can pass. Programming languagebased modeling is sometimes erroneously confused with rulebased modeling or with markup languages like SBML. The confusion with rulebased modeling arises due to the word programming because the advocators of rulebased modeling usually state that they program the behavior of biological systems through rules resembling both chemical equations and termrewriting systems. The association of programming languagebased modeling and SBMLlike formalisms arises instead due to the common word language in the two families of formalisms. SMBL is an XMLbased markup language whose overall goal is to develop an open standard that will enable simulation software to communicate and exchange models, ultimately leading to the ability for researchers to run simulations and analyses across multiple software packages (http://xml.coverpages.org/sbml.html). BlenX4Bio is not related to model exchange; it is a tabular interface suitable to biologists to program in BlenX. SBML has been originally deﬁned by merging the basic features of the formalisms supported by BioSpice [1], DBSolve [10], ECell [17], Gepasi [9], Jarnac [7], StochSim [11], and Virtual Cell [18]. Although more and more features have been added to SBML in successive versions and releases, it retains the main philosophy of chemical and mathematical modeling of the ﬁrst design thus making it philosophically diﬀerent from BlenX4Bio that instead is strongly aﬀected by the key points of Betabinders [14] and its evolution into BlenX [3] and of process algebras in general [5]. The paper is organized as follows. The next section recalls the basics of BlenX and introduces the main features of BlenX4Bio. Section 3 deﬁnes the details of the tables to be ﬁlled in for building a BlenX4Bio model by relying on examples taken from textbooks.
2
BlenX and BlenX4Bio
This section ﬁrst brieﬂy recalls the main characteristics and the inspiring principles of the programming language BlenX. Then we abstract the main parts
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of a BlenX program corresponding to the relevant information to be organized to deﬁne highlevel models of biological phenomena. The proposed interface to BlenX is based on a tabular format that resembles database tables or spreadsheet pages, both being natural tools for biologists. The organization and relationships among tables are then discussed in the BlenX4Bio paragragh. 2.1
BlenX
BlenX is a programming language based on Betabinders [14]. A program is a collection of boxes (representing biological components) with typed and dynamically varying interfaces running in parallel. The status of an interface (binder) can be available/active, hidden (for instance when a threedimensional structure changes and it hides a domain of a protein from further interactions) or complexed (when the binder is bound to the binder of another entity). Furthermore any interface is equipped with a type that describes its properties. Primitives are available to dynamically hide or unhide binders, to create new binders and to change the type of a binder to vary its properties. This feature is not available in other process calculi. The main step of computation for BlenX is the interaction between boxes or their binding/unbinding. Interaction is sensitivitybased: quantities associated to binders stochastically determine the possibility of interaction between two entities. We release the keylock interaction mechanism based on the notion, implemented in all process calculi, of exact complementarity of channel names (in fact this is an inheritance from computer science modelling where two programs can interact only if they know the exact address of the interacting partners). This allows us to avoid any global policy on the usage of names for interaction between components, i.e. we do not need centralized authorities that decide how to name entities or interfaces. BlenX supports a onetoone correspondence between biological components and boxes speciﬁed in the model: a biological component that can be in n diﬀerent states is just a box in our approach, and it is diﬀerent from mathematical/chemical modelling where n variables are needed to represent n diﬀerent states. BlenX allows users to describe complexes and their dynamic generation: BlenX users can deﬁne complexes (set of components bound together through speciﬁc interfaces) or can generate complexes during the simulation relying on the speciﬁcation of the components and on the complexation aﬃnities of their interfaces. This feature highly reduces the number of components to be speciﬁed in the initial state because the products of interactions are generated by the execution and they need not to be described initially. The high parallelism of the execution then shows all the possible scenarios. BlenX modeling improves over other processcalculi based approaches and also releases the assumption of mathematical/chemical modelling that imposes the speciﬁcations of all the species, complexes and their states reached throughout all the simulation (variables) already in the initial state (set of equations). BlenX can also specify the dynamics of systems through events: global conditions can be expressed by the amount of components or complexes in the system
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at a given time, by the simulation time or simulation steps. Conditions trigger the enabling of actions called events that are then stochastically selected including them in the set of standard interactionenabled actions. Events are used to remove or inject entities from/into the system, to join two entities into a single one, or to split an entity in two entities. BlenX events are essential for programming perturbations of the systems triggered by conditions emerging during the simulation phase, and for observing how the overall behaviour is aﬀected by them. An example could be the knockout of a gene at a given time. Another possible use of events is to program sensitivity analysis of the system into the model driven by dynamic conditions emerging during simulation. Events are essential to develop an insilico lab. No other process calculibased tool supports events in the manner described above. 2.2
BlenX4Bio
BlenX4Bio is a highlevel, tabular interface for the BlenX programming language. A BlenX4Bio model consists of a number of tables that describe all the relevant aspects that characterise biological phenomena. The BlenX4Bio tables are Compartments, Components, Complexes, Binding Aﬃnities, Dynamics, Translocation Aﬃnities and Parameters (see the tabs in Fig. 1). The ﬁrst three tables store the static information of a biological model. The tables Binding Aﬃnities and Translocation Aﬃnities represent the capabilities of the entities populating the system to interact with each other and, respectively, to change location. This information is given in terms of binding sites so that the user does not need to specify through rules or equations all the possible interactions between components. The Dynamics table provides information for driving the evolution of the model during simulation. Dynamics of systems is either expressed through simple constrained sentences like in the narrative language [6], or directly through BlenX code. Finally, the Parameters table is used to store all the quantitative information, functions and predicates introduced in the other tables. The seven BlenX4Bio tables, although structurally diﬀerent, share a number of commonalities which we describe here. Every element of each table can be addressed uniquely through its name that is made up of an alphanumeric string deﬁned by the user and a numerical identiﬁer (reported in parentheses after the name) assigned by the system to disambiguate elements with the same name. To exemplify the naming, see the deﬁnition of the two mitochondria in Fig. 3: they can be distinguished only by their numerical identiﬁers. We stress that replication of entries is, in fact, very common in biological modelling at least to model the existence of several compartments of the same kind (e.g. several lysosomes, mitochondria, and vescicles may be part of a cellular model) or to model the ubiquity of species (e.g., some species may exist simultaneously in diﬀerent compartments). Every BlenX4Bio table contains also three textual ﬁelds to describe the element under deﬁnition, to add references to the entry and to keep modeler’s note. Furthermore, all the numerical parameters are collected into the Parameters
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Fig. 1. BlenX4Bio registers all the quantitative information into the Parameters table for future updating and manipulation
table and each of them has an associated reliability value, i.e. a measure of how much the numerical datum is trusted. All the parameters can be changed and can be referred through naming conventions. This is an important feature that allows the user to dynamically change the values of the parameters, thus implementing insilico experiments. Finally all the quantities must be equipped with a suitable unit of measure. BlenX4Bio implements integrity checks of the model under development to avoid ambiguities or inconsistencies in the set of relationships deﬁned through the entries of the tables. Furthermore, BlenX4Bio supports the generation of BlenX code that can then be simulated through the BWB tool [2]. Although the main purpose of BlenX4Bio is completely diﬀerent from the one of SBMLlike formalisms [16], we conclude this section by commenting on the main diﬀerences between the current BlenX4Bio version and the most recent
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oﬃcial version of SBML, namely: SBML level 2 version 4 release 1 (which is documented at http://sbml.org/Documents/Specifications). Since BlenX4Bio is an interface of the programming language BlenX, it naturally allows for modelling of sitessensitive reactions (i.e. binding sites, phosphorylation sites, etc. are explicitly modelled in BlenX4Bio as attributes of BlenX4Bio components and sites sensitive reactions are straightforwardly expressed referring to components’ sites). Furthermore, because of the bindersbased semantics of BlenX, with BlenX4Bio reactions can be expressed either in terms of pairs of reactants (and their sites) or in terms of pair of sites (with no reference to components). The latter modality is a unique feature of BlenX4Bio (i.e. BlenX) which, prevents the speciﬁcation of a biological model from suﬀering of the infamous combinatorial explosion issue. SBML Level 2, on the other hand, does not support modelling of components’ sites: SBML biochemical species are elementary objects (i.e. sites cannot be represented) which makes SBML 2 not suitable to model sitessensitive reactions. Even though an improved third level of SBML is under development, which comprises the so called ”Multistate multicomponent species” package designed to allow, modeling of sitessensitive reactions, at present a stable version is not yet available hence it is diﬃcult to understand how the two approaches compare. Another diﬀerence between BlenX4Bio and SBML 2 regards the ability to represent translocation of species between compartments. BlenX4Bio is designed to support ﬁnegrained modelling of actual translocation mechanisms (as they are described in biology textbooks), which is: mechanisms based on transmembrane channels and translocation enablers. With SBML 2, on the other hand, translocation can be modelled only in a very abstract manner (i.e. by reactions occurring between species located in adjacent compartments): channels based mechanisms cannot be described with SBML. We also mention the ability of BlenX4Bio to describe arbitrarily large complexes like polymers without the need of introducing an arbitrary number of species as it would be the case in SBML. Finally, the possibility of using booleanvalued predicates to deﬁne statebased dynamics of models which is available with BlenX4Bio, is also not easy to render in SBML.
3
BlenX4Bio Modeling
This section illustrates the elements of the BlenX4Bio language in detail by adopting the following schema: ﬁrst we describe the biological phenomenon of interest (e.g., the spatial architecture of biological compartments of a cell, or the transportation mechanisms driving intermembrane translocations of biochemical species) by means of speciﬁc examples, then we ﬁnd a corresponding unifying abstraction, and eventually we show how such abstractions/examples are represented in BlenX4Bio. In doing so, we provide details of the structure of every BlenX4Bio table. 3.1
Spatial Architecture of Cell Compartments
From a topological perspective a cell can be seen as an architecture of physical membranebounded locations (i.e. compartments whose boundary are determined
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by a membrane). Each compartment is a container of substances and the role it plays in regulating cell life depends on the substances that it contains. The content of compartments changes over time as a result of cell regulation, hence substances continuously migrate from one location of the cell to another in response to internal and/or external stimuli. The spatial architecture of compartments, therefore, is an essential aspect of cellular biology which calls for adequate modelling support. BlenX4Bio supports modelling of the spatial architecture of compartments via the Compartments table. We illustrate modelling of compartments architecture with BlenX4Bio by means of a concrete example: the compartments architecture of Eukaryote cells, which is described next. The architecture of eukaryotes is made of a plasma membrane of about 700 µm2 that surrounds the cell interior which is partitioned into smaller subcompartments called organelles. The interior, or lumen, of each organelle is enclosed by one or more biomembranes (globally amounting at 7000 µm2 ) and contains a unique set of proteins that characterizes the functions of the organelle together with the membranebound proteins. The largest organelle is the nucleus that contains the DNA, the RNA synthetic apparatus, and a ﬁbrous matrix. The part outside the nucleus is called cytoplasm and it contains all the other organelles. The aqueous part of the cytoplasm is called cytosol and it contains its distinctive proteins as well. The major organelles, besides the nucleus, that populate the cytoplasm are detailed below.  Endoplasmatic reticulum (ER) has the largest membrane in the cell and is important for the synthesis of lipids, membrane proteins, and secretory proteins.  The Golgi complex sorts secreted and membrane proteins.  Secretory vesicles target proteins leaving the Golgi complex.  Lysosomes are acidic organelles that contain degradative enzymes.  Mitochondria are the main sites of ATP production.  Phagosomes engulf in the cytoplasm large particles from the exterior of the cell like bacteria that are then delivered to lysosomes for degradation.  Endosomes take up soluble macromolecules from the exterior of the cell and are then delivered to lysosomes.  Peroxisomes degrade fatty acids and toxic compounds.  Chloroplasts are the apparatus where photosynthesis takes place in plant cells. Diﬀerent types of eukaryotes are distinguished by the shape of the cell and the location of their organelles. The shape of a cell is determined by its cytoskeleton, a network of three kinds of ﬁlaments that mechanically support membranes. Filaments are classiﬁed according to their size into actin ﬁlaments (or microﬁlaments), intermediate ﬁlaments, and microtubules. Actin ﬁlaments provide the cell with structural support and motility; intermediate ﬁlaments provide the cell with support for the nuclear membrane and help cell adhesion into tissues formation; microtubules provide the cell with structural support, motility and cell polarity. The overall surface of the cytoskeleton (about 94000 µm2 ) oﬀers anchors to many membranes and proteins that bind to it, and works as a scaﬀold where many reactions take place.
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Fig. 2. The architecture of an eukaryotic cell that highlights its main compartments and subcompartments
An abstract representation of the architecture of eukaryotes is depicted in Fig. 2. For more information on the architecture of cells see [8], chapter 5. The representation in BlenX4Bio of a part of the architecture in Fig. 2 is reported in Fig. 3. The Compartments table always contains (even when not displayed) a global compartment termed System, which encloses all the other compartments. As a consequence, a BlenX4Bio model with no userdeﬁned compartment is perfectly legal. Besides the columns for the textual description of references to the entry, and for the user’s notes, the Compartments table contains the ﬁelds described below. The column Membrane is a tag used to distinguish between twodimensional (like, e.g., both the plasma membrane and the nuclear membrane in Fig. 2) and threedimensional compartments (all the other compartments in the same ﬁgure).
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Fig. 3. BlenX4Bio partial architecture of an eukaryotic cell
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To represent the hierarchical structure of compartments, for each newly introduced compartment, the user has to specify in which other compartment it is contained. This is the purpose of the Enclosed In column that oﬀers a selection among the already deﬁned compartments. The Size of compartments is useful to compute the kinetic rates of reactions and hence to correctly manage simulations. As for all the other quantities, for the values inserted in the Size column it is necessary to specify an accompaining reliability measure. The Quantity column denotes the number of available replicas of the same compartment. For instance, the system described in Fig. 3, contains two mitochondria. The components populating compartments can move from one compartment to another. This process is called translocation and it is driven by facilitator complexes and proteins that control transmembrane movement of entities as well as protein sorting. Such macromolecules are classiﬁed in BlenX4Bio as Translocation Enablers. An example is Importin that translocates cargo proteins from the cytoplasm to the nucleus through the nuclear pore complex. The modeling of translocation scenarios is fully described below in Section 3.5. 3.2
Components
The components that are of interest for modeling a biological system constitute the list of elementary parts from which we start simulating and analyzing the system behavior. Cells are mainly a soup of molecules varying in size and functions ﬂoating in a waterbased solution. Basic ingredients of the cell range from small molecules like ATP that stores chemical energy, hormones or neurotransmitters that transport signals, and monomers that are the basic bricks used to assemble larger molecules called polymers. The major class of polymers produced by a cell ([8], chapter 1) are: polysaccharides (whose forming monomers are sugars), proteins (whose forming monomers are aminoacids), and nucleic acids (whose forming monomers are called nucleotides). In this presentation we take elementary molecules as the smallest entities of our model, namely we work at a level of abstraction where all the possible constituents of molecules are irrelevant and hence intentionally neglected. In the same way, if we had to consider protein interactions, we would not model the aminoacids forming those proteins. Also, if protein binding and unbinding were relevant events for the considered model, we would describe either the single proteins forming the macromolecule or the complex as elementary components. We exemplify here how to completely describe the relevant information of components in BlenX4Bio by describing the constituents of a signal recognition particle (SRP) represented in Fig. 4. Each component has a unique name as all the elements of the model, and every component is characterized by a list of sites that are used to interact, bind/unbind with the other components and complexes in the model. For instance, the Importin component has a site called CargoReceptor that is used to bind other proteins. The state of such site is
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Fig. 4. Representation of elementary components in BlenX4Bio
free, meaning that it is available for binding. More generally, the current state of a site ranges over a set containing, e.g., bound, phosphorylated, sumoylated, acetylated. Another relevant information is the location where the component can be found, which can be used for managing the spatial structure of the system as well as the movement of objects. For example, Importin is located in the Cytoplasm. Sites deserve some more discussion. Proteins can freely move within compartments or can be bound to membranes spanning over both surfaces (integral proteins) or just residing on one side of the membrane (peripheral proteins) ([8],
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Fig. 5. BlenXcode corresponding to the BlenX4Bio description in Fig. 4
chapter 5). The single critical case to determine the location of a site (whether it is exposed either towards the outside of the membrane or towards the inside of the membrane) is relative to integral proteins. Indeed the exposition of the sites of integral proteins determines the set of components that can come in touch with the site due to the physical boundaries originated by the spatial hierarchical structure of cells. An example is the integral transmembrane protein P in Fig. 6 that has three sites: one directed towards the outside of the membrane (S3) and two directed towards the inside of the membrane (S1 and S2). To correctly model these situations, we distinguished compartments into membranes (2D compartments) and normal ones (3D compartments). When a component is located in a compartment that turns out to be a membrane, the site description must be extended by a further ﬁeld that says whether the site exposition is Out (towards the exterior of the membrane), In (towards the inner compartments delimited by the membrane) or Within (inside the membrane). For example Fig. 7 reports the description of the integral protein P drawn in Fig. 6. We end this subsection by noting that the information stored in the Components table is almost in a onetoone correspondence with the deﬁnition of the boxes in the BlenX program. Each component is mapped into a box and each site of a component is mapped into a corresponding interaction interface on the corresponding box. For example, the following portion of text
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Fig. 6. Binding sites of transmembrane proteins
Fig. 7. Managing locations of sites in BlenX4Bio
let P19: bproc = #(id_3, P19P54),#(id_20, P19aSRP),#(id_21, P19bSRP)[ nil ] in Fig. 5 corresponds to the description of P19 in Fig. 4. Indeed, the text is the BlenX declaration of a box named P19 with three interaction sites delimited within each #(...) and typed by P19P54, P19aSRP, and by P19bSRP, respectively.
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39
Complexes
Complexes are macromolecules formed by sets of entities bound together. Proteins can form complexes to accomplish speciﬁc functions. For instance the SRP that identiﬁes the signal sequence exiting ribosomes to translocate the synthesizing protein to the ER is made up of a piece of RNA and the six proteins P9, P14, P19, P54, P68 and P72. Protein P54 binds to the signal sequence of the synthesizing protein; P9 and P14 interact with the ribosome; P68 and P72 are required for the translocation of the protein (see [8], chapter 16 and Fig. 12). All the proteins bind directly to the piece of RNA except for P54 that binds to P19. Furthermore, P68 and P72 are a dimer as well as P9 and P14. Complexes are usually formed by bonds between domains of proteins with a high level of aﬃnity/sensitivity. A suitable abstract representation for complexes is a graph whose nodes are the boxes we used as abstract representation of components and whose arcs represent the bonds connecting the aﬃne domains of the elementary components. An abstract representation of SRP is reported in Fig. 8.
Fig. 8. Abstract representation of complexes through graphs of elementary components. The nodes represents te components and the arcs the bonds between components. Arcs have a type describing the aﬃnity between the domains or binding sites that are connected.
To uniquely refer to a bond, it is enough to assign unique names to the involved components and to the binding sites (domains) of those components. In this way we can uniquely identify a bond as, e.g., the string “component1, site1; component2, site2”. For instance if we name the RNA binding sites in Fig. 8, from left to right, SRPP19a, SRPP19b, SRPP68, SRPP72, SRPP9 and SRPP14; their complementary binding sites on the protein domains, from left to right, P19aSRP, P19bSRP, P68SRP, P72SRP, P9SRP and P14SRP;
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the binding sites that form the bond between P19 and P54, from bottom to top, P19P54 and P54P19; the binding sites that form the bond between P68 and P72, from left to right, P68P72 and P72P68; the binding sites that form the bond between P9 and P14, from left to right, P9P14 and P14P9; then the column Constituentsin Fig. 9 completely describes the SRP complex. For instance, the ﬁrst line of the table describes the binding of P14 and SRP RNA through the sites P14SRP and SRPP14. The further information stored in the Complexestable deﬁnes the quantity of the complex available at the beginning of the simulation (in this case 25 SRP) associated with a reliability measure as for all the quantitative data. As a ﬁnal observation on the Complexestable, we note that the information stored in its ﬁelds is needed to deﬁne the initial environment deﬁning the complexes that populates BlenX programs in the starting state. 3.4
Binding Aﬃnities
The chemical interactions occurring in a cell are the essence of life: they implement genetic information handling, regulation, sensing of the environment and signaling, energy production and conversion, synthesis of the material used to continuously build cells and cells’ components. Chemical interactions happen by physically binding and unbinding molecules of diﬀerent size and structure as long as they have subparts that are suitable for covalent or noncovalent bonds. A main activity of the cell involves proteins with complementary shapes and chemical properties that glue together to form complexes. Covalent bonds usually hold atoms or aminoacids together, while noncovalent bonds help deﬁning the structure of the complex. It is not only proteins to participate in binding/unbinding actions; some small molecules can serve as facilitators or building blocks for composing larger structures ([8], chapter 2). The compatibility of molecules that enable their binding is quite diﬀerent in biology with respect to the keylock complementarity that is typically used in process algebras where, for instance, to enable a communication between processes both parties must share exactly the same communication channel with no possibility of approximation. Biological shapes and chemical properties allow instead interactions between molecules that are not perfectly complementary to each other, changing the strength of the bond or the probability of the interaction. We call this notion aﬃnity, and model it in the BlenX4Bio Binding Aﬃnities table (see Fig. 10 for an overview of its ﬁelds). The sites (interfaces) through which an entity is potentially able to interact have been already identiﬁed in the deﬁnition of the corresponding entry in the Componentstable. Here, we only deﬁne aﬃnities between sites. Each site is uniquely identiﬁed by the name of the component it belongs to, and by its proper the name. We list here pairs of sites and we associate to each pair a binding rate quantifying the aﬃnity and an unbinding rate quantifying the reversibility of the binding. These quantities are then used to drive simulations. We can further detail the binding/unbinding actions by specifying conditions that must be satisﬁed in order for the action to occur. For instance, P14 and
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Fig. 9. Representation of the SRP complex in BlenX4Bio through the complete listing of the bonds identiﬁed by the names of the components, and of the sites that bind together. The four columns under Constituents refer, respectively, to the ﬁrst component, the binding site of the ﬁrst component, the second component, and the binding site of the second component.
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Fig. 10. Representation of the binding aﬃnities in BlenX4Bio needed to build the SRP complex
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Fig. 11. Representation of the binding/unbinding conditions and of the modiﬁers
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Fig. 12. The three boxes show the role of traslocation enablers in protein sorting. (A) The enablers are SRP and its receptor; (B) the enablers are the translocon and the signal region of the protein; (C) the enablers are the Import receptor and the signal region of the protein. In all the three cases, the activation of the translocation enablers by their mutual interaction open a channel in a membrane that allows one protein at time to move.
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SRP RNA can bind only if the P14SRP site of P14 is free. Similarly, we can associate modiﬁers to binding/unbinding actions to manipulate the state of the sites as a consequence of binding/unbinding. Modiﬁers can also create or delete new sites (see Fig. 11). As an example, the binding of P14 and SRP RNA creates a new free site New Site that is deleted when P14 and SRP RNA unbind. We end this subsection by noting that the information stored in the Binding Aﬃnitiestable is almost in a onetoone correspondence with the aﬃnities of the interaction interfaces of the boxes populating BlenX programs. Furthermore, binding modiﬁers and unbinding modiﬁers describe the actions that the binding/unbinding components have to take as a consequence of the formation/deletion of bonds. 3.5
Translocation Aﬃnities
Half of the kinds of proteins of a mammalian cell have to be delivered to speciﬁc locations in order for the cell to function properly. The process of moving proteins around is usually called protein targeting or protein sorting. Protein sorting can mainly happen through secretory pathways or through nonsecretory pathways ([8], chapter 16). Protein sorting is graphically illustrated in Fig. 12, where the uppermost two cartoons (A and B) show the use of secretory pathways. The secretory pathway initially targets secretory proteins to the endoplasmatic reticulum (ER)
Fig. 13. Membrane cargoreceptors recruits the proteins to be transported by a vesicle. Once the recruitment is complete, the membrane forms a vesicle with a Rab+GTP receptor for docking the vesicle towards the right target membrane. Once the rab eﬀector on the target membrane is identiﬁed and bound, the two SNARE counterparts tightly bind together allowing the fuse of the vesicle with the target membrane. As a consequence, the whole content of the vesicle is delivered in the compartment bound by the target membrane. The translocation enablers here are the vSNARE and tSNARE proteins.
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membrane and translocated in the ER lumen (see Fig. 12 A and B). When translation is completed these proteins are further delivered to the lumen of Golgi and lysosomes or to plasma membranes via vesicles. A uniform principle drives this type of protein sorting: proteins are moved from one membranebounded compartment to another membranebounded compartment mediated by transport vesicles ([8], chapter 17). Vesicles collects cargo proteins in buds formed from the membrane of the source compartment and then fuse with the membrane of the target compartment by releasing in one shot all their content (see Fig. 13). The nonsecretory pathway sorts proteins to the ER, mitochondria (see Fig. 12 C), chloroplasts, peorxisomas, and the nucleus. The main diﬀerence with respect to the situation described above is that these proteins are not secretory proteins and most of them are completely translated into the cytoplasm before their translocation. The mechanism of translocation is similar for all the cases mentioned so far apart from nuclear import/export ([8], chapter 12.3). Nuclear transport happens through the nuclear pore complexes (NPC) that form holes on the nuclear membrane. Macromolecules equipped with nuclear localization signals (NLS) bind to importin/exportin proteins that drive the macromolecules through the nuclear pores (see Fig. 14). We classify all the above kinds of translocation phenomena into a single abstract translocation process suitable for modeling and mapping into the BlenX language. The abstract translocation phenomenon can be seen as a door that
Fig. 14. A free importin binds to a cargo protein in the cytoplasm and the complex translocates into the nucleoplasm through the nuclear pore. The importin part of the complex interacts with RanGTP that causes a decrement of the aﬃnity between the importin and the cargo protein. As a consequence the cargo protein is released in the nucleoplasm and the free importin migrates back to the cytoplasm. Similarly, cargo proteins can migrate from the nucleoplasm to the cytoplasm by forming a complex with an exportin and a RanGTP. The translocation enablers are here the importin and exportin proteins.
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Fig. 15. BlenX4Bio model of translocations
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can be opened by unlocking it simultaneously with two diﬀerent keys. In the case of the nucleus the second enabler can coincide with the cargo protein that must complex with importin to enter the nucleus or with exportin to exit the nucleus. Once the door is open (e.g., nuclear pore, translocon Sec61, general import pores Tom40, Tim23/17, fuse of membranes in vesicular cases), its size determines how many entities can simultaneously step through the door (the throughput of the translocation channel). We distinguish the vesicular case in which all the content is translocated simultaneously from all the other cases discussed where a single entity at time can pass. The BlenX4Bio table Translocation Aﬃnities is the place where the above abstract representation of translocation phenomena is stored. For instance, Fig. 15 shows how the enablers of the protein sorting to ER (Fig. 12A) are modeled. The Translocation Aﬃnities table speciﬁes the components acting as enablers and some quantitative information on translocation. The enablers reported in Fig. 15 are the SRP in the cytoplasm and AlfaBeta, an integral protein of the ER membrane for protein sorting to ER; vSNARE and tSNARE for vesicular fusion with other compartments; Importin and the nuclear pore complex (NPC) to import cargo proteins into the nucleus; and ﬁnally Exportin and NPC to export cargo proteins from the nucleus. We further specify the rate of the translocation and its throughput, i.e. the speed and the bandwidth of the channel that allows translocation of components expressed in numbers of components per unit of time that traverse the channel. Finally we specify a condition that enables the opening of the channel and hence favours translocation. 3.6
Dynamics
So far we described how to represent the structural components of the system under investigation by deﬁning the hierarchical space architecture and by listing the basic entities and complexes available in the initial state which is the starting point of the simulation. Furthermore, information on aﬃnities for binding has been provided. Below, we discuss how the system behaves when time passes and how the simulation process deals with the static information deﬁned in the tables already described. The dynamics of systems is speciﬁed in the BlenX4Bio Dynamics table. All the entries have a unique name for referral purposes and are characterized by an algorithmic description of the action to be taken in order to make the system evolve. These descriptions are characterized by a conditional part and an action part that is enabled only when the side condition is satisﬁed. The expressions stating the dynamics of an event are speciﬁed in a constrained natural language similar to the narrative language [6]. Examples are reported in Fig. 16. Note that the dot notation is used to identify subparts of components. For instance, AlfaBeta.Alfa refers to the site Alfa of the component AlfaBeta (we do not describe here the full language because it is outside the scope of the present paper and we refer to the operative documentation of BlenX4Bio). Since the dynamics is quantitative and stochastic in nature, we must associate every entry of the Dynamics table with a rate.
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Fig. 16. BlenX4Bio model of system dynamics
We observe that the information stored in the Dynamics table deﬁnes the algorithm to be executed to simulate the temporal evolution of the described biological systems. This is the kernel of the BlenX program corresponding to the model speciﬁed through BlenX4Bio, and indeed the user can choose to specify the dynamics by directly writing BlenX code.
4
Conclusions and Future Work
The paper presented a tabular interface to the BlenX language that hides all the programming details from the users of BlenX4Bio. The CoSBi Lab platform supports the translation of BlenX4Bio models into BlenX programs ready to be simulated. In order to demonstrate the soundness of BlenX4Bio representation of biology we have considered basic aspects of cellular biology taken from biology textbooks and we provided evidence that BlenX4Bio is indeed a very useful means through which biologists can straightforwardly represent biological phenomena in a simple and intuitive manner. The key characteristics that distinguish BlenX4Bio from SBML or equational/chemical modeling have also been discussed. We are currently working on extensions of the BlenX4Bio modelling approach. In fact, if on one hand BlenX4Bio allows the modeler to specify the biological system to be studied, on the other hand the actual analysis of the modeled systems requires some additional information to be provided. We envisage such additional information consisting of two parts: the aspects of the system’s
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behaviour which is relevant to look at and the ways in which the speciﬁed model could be perturbed. In our extension of BlenX4Bio, experiments describe in a formal manner which are the parts of the model that the modeler wishes to perturb in order to analyse the behaviour of the modelled system. Essentially, a BlenX4Bio experiment consists of two elements: a description of the (possibly conditional) perturbations the model is subject to during the experiment, and a description of the relevant behavioural characteristics (expressed in terms of queries) the modeller wishes to concentrate on in order to analyse the modelled system. We are currently extending the CoSBi Lab software to support the experiments speciﬁcation extension of BlenX4Bio and hence move towards a fully integrated modeling and insilico experimental platform for biologists. Acknowledgements. The authors thank A. CsikaszNagy, L. Dematt´e, M. Forlin, M.L. Guerriero, J. Heath, A. Ihekwaba, A. Inga, R. Larcher, P. Lecca, R. Mardare, T. Mazza, I. Mura, A. Palmisano, N. Phuong, D. Prandi, A. Quattrone, A. Romanel, and J. Zamborsky for useful discussions. The authors thank F. Benedetti and the CoSBi development team for the implementation of BlenX4Bio.
References 1. BioSpice home page, http://biospice.sourceforge.net/ 2. Dematt´e, L., Priami, C., Romanel, A.: The Beta Workbench: a computational tool to study the dynamics of biological systems. Brief. Bioinform. 9(5), 437–449 (2008) 3. Dematt´e, L., Priami, C., Romanel, A.: The BlenX Language: A Tutorial. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 313–365. Springer, Heidelberg (2008) 4. Guerriero, M.L., Dudka, A., UnderhillDay, N., Heath, J.K., Priami, C.: Narrativebased computational modelling of the gp130/jak/stat signalling pathway. BMC Systems Biology 3(40) (2009) 5. Guerriero, M.L., Prandi, D., Priami, C., Quaglia, P.: Process calculi abstractions for biology. In: Condon, A., Harel, D., Kok, J.N., Salomaa, A., Winfree, E. (eds.) Algorithmic Bioprocesses. Springer, Heidelberg (2009) 6. Heath, J.K., Guerriero, M.L., Priami, C.: An automated translation from a narrative language for biological modelling into process algebra. In: Calder, M., Gilmore, S. (eds.) CMSB 2007. LNCS (LNBI), vol. 4695, pp. 136–151. Springer, Heidelberg (2007) 7. Jarnac home page, http://www.sysbio.org/software/jarnac.htm 8. Lodish, H., Berk, A., Matsudaira, P., Kaiser, C.A., Krieger, M., Scott, M.P., Zipursky, S.L., Darnell, J.: Molecular Cell Biology, 5th edn. Freeman, New York (2004) 9. Mendes, P.: Biochemistry by numbers: simulation of biochemical pathways with Gepasi 3. Trends Biochem. Sci. 22, 361–363 (1997) 10. Moehren, G., Markevich, N., Demin, O., Kiyatkin, A., Goryanin, I., Hoek, J., Kholodenko, B.N.: Temperature dependence of epidermal growth factor receptor signaling network can be accounted for using a kinetic model. Biochemistry 41(1), 306–320 (2002)
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11. Le Novere, N., Shimizu, T.S.: StochSim: modelling of stochastic biomolecular processes. Bioinformatics 17, 575–576 (2001) 12. Priami, C.: Algorithmic systems biology. CACM 52(5), 80–88 (2009) 13. Priami, C., Quaglia, P.: Modeling the dynamics of biosystems. Brieﬁngs in Bionformatics 5(3) (2004) 14. Priami, C., Quaglia, P.: Beta Binders for Biological Interactions. In: Danos, V., Sch¨ achter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 20–33. Springer, Heidelberg (2005) 15. Regev, A., Shapiro, E.: Cells as computations. Nature 419, 343 (2002) 16. SBML home page, http://sbml.org/ 17. Tomita, M., Hashimoto, K., Takahashi, K., Shimizu, T.S., Matsuzaki, Y., Miyoshi, F., Saito, K., Tanida, S., Yugi, K., Venter, J.C., Hutchison, C.A.3.: ECELL: software environment for wholecell simulation. Bioinformatics 15(1), 72–84 (1999) 18. Virtual Cell home page, http://www.ibiblio.org/virtualcell/
Modelling Biological Clocks with BioPEPA: Stochasticity and Robustness for the Neurospora crassa Circadian Network Ozgur E. Akman1 , Federica Ciocchetta2 , Andrea Degasperi3, and Maria Luisa Guerriero4 1
3
Centre for Systems Biology at Edinburgh, The University of Edinburgh, Edinburgh EH9 3JU, Scotland, UK 2 The Microsoft Research  University of Trento Centre for Computational and Systems Biology, Trento, Italy Laboratory for Foundations of Computer Science, The University of Edinburgh, Edinburgh EH8 9AB, Scotland, UK 4 Department of Computing Science, The University of Glasgow, Glasgow G12 8QQ, Scotland, UK
Abstract. Circadian clocks are biochemical networks, present in nearly all living organisms, whose function is to regulate the expression of specific mRNAs and proteins to synchronise rhythms of metabolism, physiology and behaviour to the 24 hour day/night cycle. Because of their experimental tractability and biological significance, circadian clocks have been the subject of a number of computational modelling studies. In this study we focus on the simple circadian clock of the fungus Neurospora crassa. We use the BioPEPA process algebra to develop both a stochastic and a deterministic model of the system. The light on/oﬀ mechanism responsible for entrainment to the day/night cycle is expressed using discrete timedependent events in BioPEPA. In order to validate our model, we compare it against the results of previous work which demonstrated that the deterministic model is in agreement with experimental data. Here we investigate the eﬀect of stochasticity on the robustness of the clock’s function in biological timing. In particular, we focus on the variations in the phase and amplitude of oscillations in circadian proteins with respect to diﬀerent factors such as the presence/absence of a positive feedback loop, and the presence/absence of light. The timedependent sensitivity of the model with respect to some key kinetic parameters is also investigated.
1 Introduction Circadian clocks are oscillatory gene networks developed by living organisms in order to adapt to the 24hour day/night cycle. In general, the biochemical mechanisms regulating circadian rhythms are robust enough for approximately 24 hour oscillations to persist over a range of constant lighting and temperature conditions. Exposure to periodic external stimuli (e.g. light/dark or temperature cycles) has the eﬀect of resetting these freerunning oscillations so as to establish stable phase relationships with the forcing stimulus (circadian entrainment). This enables cyclic changes in the environment to P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 52–67, 2009. c SpringerVerlag Berlin Heidelberg 2009
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be anticipated, such as seasonal variations in the length of day (photoperiod) [1]. Circadian rhythms are present in nearly all eukaryotes, from mammals and plants, to insects and fungi. There is now detailed experimental data showing that these rhythms can be produced by networks of multiple, interlocked positive and negative feedback loops in which the protein product of a gene modulates expression of either its own transcript or that of another target gene in the network [2]. Several mathematical models have been proposed in recent years to describe the specific oscillationgenerating mechanisms in a range of diﬀerent organisms. These include the fruit fly Drosophila melanogaster [3,4], the plant Arabidopsis thaliana [5,6] and the mouse Mus musculus [7,8]. Here we focus on the fungus Neurospora crassa, which possesses one of the most comprehensively studied circadian networks [9]. In recent years, a number of mathematical models of the Neurospora clock have been developed, including continuousdeterministic models that are described in terms of ordinary diﬀerential equations (ODEs) [10,11,12,13,14], as well as discretestochastic models [15,16]. Such models have been used successfully to explore the relationship between the architecture of the Neurospora circadian network and the robustness of its function in biological timing. Within this theme, our aim in this work is to investigate the eﬀect of stochastic fluctuations on the performance of the Neurospora clock. While deterministic models are good approximations of real biochemical systems when the number of molecules is suﬃciently high, at low copy numbers the eﬀect of random fluctuations becomes significant and so stochasticity needs to be taken into account to obtain a faithful representation of the real biochemical system [17]. To explore the eﬀect of these fluctuations on circadian timing in Neurospora, we implement a discretestochastic version of a continuous ODE model previously developed to investigate the entrainment of the clock by light and temperature [13,18]. We use the ODE representation of this clock to validate our stochastic model and to highlight the diﬀerences between deterministic and stochastic representations of the network. In particular, where previous stochastic studies have concentrated mainly on the unforced (freerunning) Neurospora clock, modelling entrainment as a weak modulation of transcription [15,16], here we investigate how stochasticity aﬀects the robustness of circadian oscillations for a more realistic model which explicitly incorporates elements of the lightsignalling pathway [13,18]. We exploit discrete timedependent events to represent light/dark cycles and analyse the behaviour of the system under diﬀerent light conditions and in the absence of a core feedback loop. As part of this analysis, we use a novel sensitivity analysis method to determine the time within the circadian cycle at which a given phase marker is most responsive to parameter variations. We use BioPEPA [19,20] as our modelling language. BioPEPA is a process algebra recently developed for modelling biochemical systems. Among its key strengths as a language for systems biology is the fact that it is equipped with diﬀerent semantics, enabling both continuousdeterministic and discretestochastic representations of the same model description to be automatically generated. Another important feature of BioPEPA is that it permits the definition of generic rate laws. This allows the specification of complex kinetic formulae, such as those used in the ODE representation of the original Neurospora model (see Sect. 4 below). In addition, timedependent events
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can be easily incorporated, enabling periodic external stimuli such as light/dark cycles to be represented in a straightforward manner. The rest of the paper is structured as follows. The circadian clock of Neurospora crassa and the BioPEPA model of the clock are described in Sect. 2 and Sect. 4, respectively. BioPEPA is introduced in Sect. 3. In Sect. 5 the simulation and analysis results are presented. Finally, in Sect. 6 we report some concluding remarks.
2 The Circadian Clock of Neurospora Crassa Neurospora exhibits a 22 hour rhythm in asexual spore formation (conidiation) when grown in constant darkness (DD). The conidiation rhythm is a key clock output which can be entrained by both light and temperature [21]. In natural 24 hour cycles of alternating light and dark (LD), the phase of entrainment (judged by the time of conidiation onset) coincides with the middle of the night in both long and short days, providing a simple, biologically relevant measure of circadian function [22,13,18]. The core, multiloop genetic oscillator believed to underlie the conidiation rhythm is formed by the rhythmic gene frequency (frq) and the constitutively expressed gene white collar1 (wc1) [9]. The protein product of the white collar1 gene, WC1, comprises the positive element of a central negative feedback loop, activating transcription of frq. The protein product of the frq gene, FRQ, is the negative element of the loop, interacting with frqbound WC1 to inhibit frq expression [23,24]. In addition to its role as a transcriptional inhibitor, FRQ positively regulates expression of WC1, giving a positive feedback loop that interlocks with the central loop [25]. Light entrains the clock by promoting the binding of a flavin chromophore to WC1, resulting in a lightactivated form which enhances frq transcription [24]. A network diagram for the model of the core oscillator that we consider here is shown in Fig. 1. For the ODE representation of the model presented in [13,18], the repressive
Fig. 1. A schematic representation of the gene network underlying the model of the Neurospora clock. WC1* represents lightactivated WC1. The dashed lines indicate lightdependent geneprotein interactions.
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action of FRQ transcription factor (hereafter called PF) on frqbound WC1 transcription factor (PW) and frqbound lightactivated WC1 transcription factor (PWL) was assumed to occur through a noncompetitive inhibition process modelled using Hill kinetics. Hill kinetics were also used to describe the upregulation of WC1 translation by PF as well as the lightdependent increase in the transcription of wc1 mRNA (MW), necessary to simulate lossoffunction wc1 mutants [13]. MichaelisMenten kinetics were used to describe enzymemediated degradation of mRNA and protein, while the conversion of PW to PWL was modelled as a reversible first order massaction reaction PW PWL with a lightdependent forward rate. The light input to the ODEs took the form of a smoothly diﬀerentiable function that switches rapidly between 0 and 1 at dawn (t = tdawn ), and from 1 back to 0 at dusk (t = tdusk ), modelling the lighting protocol commonly used in circadian experiments. In order to obtain oscillatory behaviour, a delay was introduced into the central negative feedback loop by assuming that justtranslated FRQ protein (E1F) is modified into a second intermediate protein (E2F) before being converted into transcription factor [13]. The conversion processes MF → E1F → E2F → PF, which include translation of FRQ from frq mRNA (MF), were each modelled as first order massaction reactions. Similarly, a delay was introduced into the positive feedback loop by introducing two intermediate WC1 protein species (E1W and E2W), and describing the conversions MW → E1W → E2W → PW with first order kinetics. The ODE representation of the model comprises 9 equations with 34 kinetic parameters. The parameters were fitted to gene and protein expression time series in DD and LD using a bipartite optimisation method developed for highdimensional computational biology models [5]. This technique combines a random parameter search with simulated annealing to minimise a qualitative cost function that assesses the goodnessoffit of the model to key experimental data sets [5,13]. For the ODE model, the best parameter set was taken to be that yielding the smallest cost function score following the application of the optimisation scheme to 50 million randomly distributed points in the 34dimensional parameter space [18]. This optimal parameter set yielded a good fit to each of the target time series, and also reproduced the variation in entrainment phase with photoperiod observed experimentally [18]. We use the same parameter set here for the BioPEPA representation of the model detailed in Sect. 4.
3 BioPEPA In this section we give a short description of BioPEPA [19,20], a language that has recently been developed for the modelling and analysis of biological systems. The main components of a BioPEPA system are the species components, describing the behaviour of each species, and the model component, describing the interactions between the species and initial amounts. The syntax of the BioPEPA components is defined as: S ::= (α, κ) op S  S + S  C
with op = ↓  ↑  ⊕  
P  S (x) P ::= P L
where S is the species component and P is the model component. In the prefix term (α, κ) op S , κ is the stoichiometry coeﬃcient of species S in reaction α, and the prefix
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combinator “op” represents the role of S in the reaction. Specifically, ↓ indicates a reactant, ↑ a product, ⊕ an activator, an inhibitor and a generic modifier. We can use the shorthand notations (α, κ) op and α op for (α, κ) op S and (α, 1) op S , respectively. The operator “+” expresses the choice between possible actions, and the constant C def Q denotes synchronisation between is defined by an equation C = S . The process P L components P and Q; the set L determines those activities on which the operands are forced to synchronise, with denoting a synchronisation on all common action types. ∗ In the model component S (x), the parameter x ∈ R represents the initial concentration (or the number of molecules in a discretestochastic setting). The reader is referred to [19] for further details on the language and its semantics. Recently BioPEPA has been extended to incorporate events [26], constructs that represent changes in the system due to some triggering conditions. This allows biochemical perturbations to the system to be represented, such as the timed introduction of reagents or the modulation of system components by external stimuli. A BioPEPA event has the form (id, trigger, event assignment, delay), where id is the event name, trigger is a mathematical expression involving the components of the BioPEPA model and/or time, event assignment is a list of assignments causing some changes to elements in the system, and delay is either 0 (immediate events) or a positive real value (delayed events). A BioPEPA system representing a biochemical network consists of a set of sequential components, a model component, and context (defining information such as kinetics rates, parameters, locations, and events). Its formal definition is the following: Definition 1. A BioPEPA system P is a 8tuple t, L, N, K, FR , Comp, P, Events , where: t is time, L is the set of locations, N is the set of (auxiliary) information for the species, K is the set of parameters, FR is the set of functional rates, Comp is the set of species components, P is the model component and Events is the set of events. BioPEPA oﬀers a formal representation of biochemical systems, on which diﬀerent kinds of analysis can be carried out, through the defined mappings into continuousdeterministic and discretestochastic modelling languages. The BioPEPA language is supported by software tools which automatically process BioPEPA models and generate other representations in forms suitable for diﬀerent kinds of analysis [19,27]. In particular, the generated simulation model can be executed using the Dizzy simulation tool [28], in which both stochastic simulation algorithms and diﬀerential equation solvers are implemented.
4 The BioPEPA Model of the Circadian Clock In the following we provide an overview of the BioPEPA model for the circadian clock described in Sect. 2. The full model is reported in the Supplementary Material. The clock is characterised by robust entrainment to light/dark cycles. Light entrains the clock by modulating general kinetic laws diﬀerent from massaction that abstract complex sequences of more elementary steps [13,18]. These features can be easily represented in BioPEPA using events and functional rates. From the BioPEPA description of the clock we can derive both the model for stochastic simulation and the related system of diﬀerential equations.
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In order to derive a stochastic model, the continuous concentration values of ODE models must be translated into discrete numbers of molecules. In general, assuming concentrations are expressed in molars (M), the initial amounts must be multiplied by the factor NA · V (where NA is the Avogadro number and V is the volume of the compartment in which the reactions take place), and the kinetic parameters must be rescaled accordingly (see [29] for details). For the Neurospora model, as the compartment size and absolute concentration values are not known to any great accuracy, we follow the approach used in [15] and introduce a generic scaling factor Ω that modulates the number of molecules. Specifically, concentrations are turned into discrete numbers of molecules by multiplying them by Ω, and the kinetic parameters are also rescaled by Ω (see the Supplementary Material for details). Reactions and kinetic laws. Each reaction is associated with an action type and a functional rate, expressing the kinetic law. For instance, the transcription of MF under upregulation by PWL and inhibition by PF is represented by the action type transcription MF by PWL and the kinetic law: a1 · PWLn (1 + (PF/b1 )g ) · (PWLn + b2 n ) Species. Each biological species is abstracted by a BioPEPA species component. Below we report the definition of PF; the other species are described similarly. transcription MF by PWL =
def
PF = (transcription MF by PW, 1) + (transcription MF by PWL, 1) + (transformation E2F to PF, 1) ↑ + (degradation PF, 1) ↓ + (translation E1W by PF, 1) ⊕ PF is involved in five reactions: it is an inhibitor of the transcription of MF with and without the influence of light (first line), a product of the transformation from E2F to PF, a reactant in the degradation of PF (second line) and an activator of the translation of E1W (last line). Note the use of shorthand notation in the definition of PF. The full system is described in terms of the model component
MF(m f 0) E1F(e1 f 0) E2F(e2 f 0) PF(p f 0) MW(mw0) ∗ ∗ ∗ ∗ ∗
PW(pw0) PWL(pwl0) E1W(e1w0) E2W(e2w0) ∗ ∗ ∗
where the values in parenthesis are the initial values for the species. Events. Entrainment by light/dark cycles is represented by events in BioPEPA. In the initial state the system is in dark conditions and, therefore, the transformation from the protein PW to the form activated by light (PWL) is not possible. This is represented by setting the kinetic parameter r1 for the transformation reaction PW → PWL equal to 0. At dawn, the reaction is suddenly activated and therefore r1 is reset to its maximum value 5.1759. At dusk the reaction is deactivated again by resetting r1 to 0. This periodic sequence of parameter changes is represented by the following set of immediate events Events = [(dawni ; t = tdawn · i; r1 = 5.1759; 0), (duski ; t = tdusk · i; r1 = 0; 0), i = 1, 2, . . . , D ]
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where D is the number of simulated days, and tdawn and tdusk are the times of the day at which dawn and dusk occur, respectively. By changing the values of tdawn and tdusk we can simulate the eﬀect of changing the photoperiod (tdusk − tdawn ), a key entrainment parameter for the Neurospora clock [22,18]. Here we focus on two conditions: constant darkness (DD) and alternating 12hour cycles of light and dark (12:12 LD).
5 Model Analysis In this section we present the validation of our model against the original ODE representation and we illustrate some analysis results. We use a version of the Dizzy simulator [28] developed at the University of Edinburgh [30], which extends the tool with sensitivity analysis techniques and additional simulation methods. The timedependent events in the BioPEPA model are translated into timedependent reaction rates in the Dizzy model (defined in terms of the step function theta, which is predefined in Dizzy), and we use the GibsonBruck stochastic simulation algorithm [31]. The choice of this algorithm is due to its eﬃciency (in simulation time) with respect to other stochastic simulators and to the fact it supports timedependent rates. 5.1 Validation of the Model As a preliminary step, we validate the BioPEPA model by comparing it against the original deterministic representation [18,13]. In Fig. 2(a) and Fig. 2(b) we show the comparison for the DD system and for 12:12 LD cycles respectively. In each graph we plot three timeseries: the behaviour of the original model (dashed lines), the solution of the system of ODEs generated by the BioPEPA model (solid lines), and the average behaviour over 10 stochastic simulation runs with scaling factor Ω = 10000 (points). The variables plotted are the clock outputs frq mRNA (MF), wc1 mRNA (MW), total FRQ protein (FP = E1F + E2F + PF), and total WC1 protein (WP = E1W + E2W + PW + PWL).
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Fig. 3. The eﬀect of increasing Ω on the LD system. Plotted are timeseries showing oscillations in total FRQ protein FP averaged over 1000 stochastic simulations for Ω=10, 100 and 1000 (coloured lines). Each time series has been normalised by Ω to enable comparison with the FP oscillations obtained from the deterministic model (black line).
The scaling factor Ω = 10000 was chosen with the purpose of having a reasonably high number of molecules to minimise the eﬀect of stochastic fluctuations [29]. Consequently, in both DD and LD systems, the timeseries resulting from the deterministic and the stochastic analysis of the BioPEPA models are in very close agreement, with the stochasticity almost unnoticeable, despite the small number of simulation runs. Comparing the deterministic timeseries generated by our model against that of the original model, we observe that in the DD system they are in perfect agreement, while a small diﬀerence can be observed in the LD system (especially for WP): the reason for this diﬀerence resides in the diﬀerent ways in which the light switch is modelled: a smooth function in the original ODE model versus discrete events in the BioPEPA model. A similar agreement was obtained for diﬀerent photoperiods (results not shown). 5.2 Eﬀect of the Scaling Factor Ω on Stochasticity Higher values of the scaling factor Ω correspond to larger molecular populations in the stochastic model, yielding smaller stochastic fluctuations [29]. We have seen in the previous section that for Ω = 10000 the stochasticity is reduced to such a point that even with a small number of simulations runs, the average simulation behaviour is nearidentical to the deterministic behaviour. As a consequence, the higher the scaling factor, the more regular the circadian oscillations will be, whereas we expect the eﬀects of noise to be more evident with a smaller scaling factor. Figure 3 shows the average oscillations in total FRQ protein FP for diﬀerent values of Ω (10, 100 and 1000). We observe that for Ω = 10 the average behaviour of the stochastic system diﬀers significantly from the FP oscillation in the deterministic system, yielding unstable oscillations that are inconsistent with the stable cycling of FRQ observed experimentally [9]. By contrast, for Ω = 100 and Ω = 1000, regular oscillating dynamics are obtained. We also note that the average oscillations for Ω = 100 and Ω = 1000 are very close to the deterministic solution, indicating that increasing Ω in this range only aﬀects the variability about the average. We consider a scaling factor Ω = 1000 in the remainder of the work. Similar results (with slightly higher variability) were obtained in all cases with Ω = 100.
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5.3 Investigating the Role of Positive Feedback In this section, we study how the positive feedback loop of the Neurospora network aﬀects its stochastic behaviour. A previous analysis of the deterministic model identified the upregulation of WC1 production by FRQ transcription factor, controlled by the parameter a7 , as a key regulatory process in the light entrainment of the clock [18]. With a7 at its nominal value aWT = 2.4695, the ODE model yields the correct exper7 imental responses to changes in photoperiod, with the phase of total FRQ protein FP tracking the middle of the night in both long and short days [18]. Knocking out positive feedback by reducing a7 to 0 destroys selfsustained oscillations in DD by pushing the deterministic model through a supercritical Hopf bifurcation [18]. This is in agreement with the loss of freerunning conidiation rhythms reported in experiments [32]. The destruction of the DD limit cycle has a significant eﬀect on the light responses of the model, yielding a system that is unable to respond to changes in photoperiod during long days. This suggests a role for the positive loop in promoting robustness against seasonal photoperiod changes [18]. Here, we compare the behaviour of the stochastic and deterministic models for DD and 12:12 LD cycles in the presence and absence of the positive loop, focusing on the resulting changes to the FRQ oscillation in each case. The Eﬀect of Removing Positive Feedback on the DD System. Figure 4(a) shows the diﬀerence between the deterministic and stochastic behaviour for the unperturbed network in DD. While the ODE model exhibits selfsustained FRQ oscillations, the average oscillation generated by the stochastic system damps to a constant value. This is a consequence of the individual realisations of the stochastic model going out of phase with each other, as can be seen in Fig. 4(c) and Fig. 4(d). This phase diﬀusion in the freerunning system, characterised by a phase distribution spanning the full circadian cycle, agrees with previous stochastic analysis of circadian models [15]. Setting the positive feedback strength a7 to 0 yields damped FRQ oscillations in the ODE model, as a7 is below the Hopf bifurcation value (Fig. 4(b)). Individual realisations of the stochastic model, however, are still oscillatory, albeit with smaller amplitudes compared to the unperturbed network (Fig. 4(c)). Again, the average FRQ oscillation damps to a constant value as a consequence of phase diﬀusion. The persistence of selfsustained oscillations when positive feedback is removed demonstrates that stochasticity can introduce greater robustness against modifications to the network architecture. This finding is consistent with models of the mammalian clock for which simulated knockouts that are arrhythmic in ODE implementations can become rhythmic when stochasticity is incorporated [8]. The Eﬀect of Removing Positive Feedback on the LD System. Finally, we consider the 12:12 LD system and examine the eﬀect of setting a7 to 0 on the oscillatory behaviour of the model. Comparing Fig. 4 and Fig. 5, it is clear that for both the unperturbed system and the positive loop knockout, entrainment regularises the dynamics, markedly reducing the variability of oscillations compared to the freerunning system (similar findings were reported for a model of the Drosophila clock in [15]). In both cases, there is relatively little phase diﬀusion, as evidenced by phase distributions that are concentrated about their corresponding deterministic values (Fig. 5(c)). Interestingly, although removal of positive feedback shifts the mean value of FRQ phase,
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consistent with the analysis of the deterministic system ([18]), the variation about the mean is unaﬀected (the standard deviation is 0.3720 for the unperturbed system and 0.3238 for the loop knockout). This demonstrates that the positive loop is able to buﬀer the clock against environmental variations (seasonal changes in photoperiod) without degrading its robustness to stochastic fluctuations in the chemical reactions comprising the oscillatory mechanism. 5.4 Sensitivity Analysis for the 12:12 LD System Sensitivity Analysis (SA) aims to identify the relationships between the inputs and outputs of mathematical models of biochemical networks [33]. A key goal is the production of Sensitivity Indices (SI) that quantify these relationships, revealing which factors are the most influential with respect to model outcome. The most widespread SA method is “oneatatime” (OAT). Given a mathematical model with parameters set to those considered the most likely (also called nominal parameters), each parameter is perturbed individually by a fixed value or by a percentage of its nominal value, and the change in the output(s) of interest measured. OAT has seen widespread use in ODE models of biochemical interactions; this has included circadian networks for which a standard
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approach has been to compute the sensitivities of period and amplitude over one cycle of the oscillation [16,11,12]. In [34] this method has been extended to stochastic models. In this case, the output at a given time is not just a value representing the amount of a species as in ODEs; it is, instead, a set of possible values, obtained from independent stochastic simulations. The SA extension has been obtained by substituting the diﬀerence between perturbed and nominal output values employed in the traditional approach with a diﬀerence measure based on the density distribution surface of the output, estimated with a suitable number of simulations. An estimate of this density distance based on stochastic simulations can be obtained using histogram distance, as originally presented in [35]. This stochastic version of OAT therefore applies when one is interested in observing the change in the distribution of the amount of a particular species at a given time. Here we apply the traditional approach to the means of the stochastic simulations and also consider the extended approach, based on histogram distance. These are indeed complementary, as the former does not incorporate any notion of stochastic variability, while the latter quantifies the likelihood of having the same distribution in both the perturbed and unperturbed systems. Moreover, a feature of the histogram distance is that its value will always be 2 when there is no overlapping of the distributions, making the traditional approach still necessary to determine sensitivities for large displacements.
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Figure 6 summarises the result of the local sensitivity analysis obtained by changing a subset of parameters predicted to have a significant eﬀect on entrained phase. Each parameter was incremented by 10%, and the results shown are averages over 1000 simulation runs for Ω = 100 and Ω = 1000. It can clearly be seen that both the densityand averagebased sensitivity measures vary significantly over the circadian cycle for the parameters considered. For both measures, the most sensitive parameters are a4 , d1 and d3 representing the maximum rates of lightindependent wc1 transcription, frq degradation and wc1 degradation respectively. All 3 parameters yield maximum sensitivities with respect to the averagebased measure around dusk, when FP is close to its peak value. By contrast, maximum sensitivities with respect to the densitybased measure occur at the time when FP decreases to its halfmaximum value, a molecular correlate of conidiation onset [22,13,18]. This demonstrates that in terms of the average FRQ oscillation, the marker of entrained phase most responsive to evolutionary parameter variations is peak FRQ phase. However, when stochastic variations in the FRQ waveform are considered, the most responsive marker is the phase of the FRQ halfmaximum. As it is the latter which correlates with physiological entrained phase for the Neurospora clock, this analysis suggests that stochastic fluctuations in FRQ expression
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Fig. 6. Local sensitivity to parameter variation. Sensitivities were computed every 3 hours over one circadian cycle (72 ≤ t ≤ 96). The color gradient represents the diﬀerence in the amount of FP between the nominal and modified parameter sets in each case (with increasing sensitivity going from black to white). In all panels, the light blue lines denote the cycle phase at which maximum sensitivity is attained for a given parameter. Because of the periodic behaviour of the system, qualitatively similar sensitivities are obtained over other 24h intervals (see Fig. S1 in the Supplementary Material).
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may have been an important contributing factor to the selection of the halfmaximum as a phase marker in nature. As previously mentioned, averagebased SA computes diﬀerences in terms of the average FP value, while densitybased SA also considers the stochastic variability. Therefore, the diﬀerences in the sensitivities when using diﬀerent scaling factors can only be captured consistently by using densitybased SA. For instance, similar averagebased sensitivities are obtained for Ω = 100 and Ω = 1000 (cf. Figs. 6(a) and 6(c)), while the densitybased sensitivities are in general smaller for Ω = 100 as a consequence of the higher variance causing greater overlapping in the probability distributions of FP (cf. Figs. 6(b) and 6(d)).
6 Conclusions In this work we presented and studied a stochastic model of the circadian clock in Neurospora crassa under two diﬀerent light conditions: constant darkness and 12:12 light/dark cycles. We used BioPEPA as our modelling language. This language allowed us to represent in a straightforward manner two features of the system: complex kinetic laws and timedependent events representing cyclic light/dark conditions. The model was validated against an existing ODE representation describing key behaviours observed in laboratory experiments, including the variation of entrained phase with photoperiod. We presented some analysis results illustrating the diﬀerences between deterministic and stochastic representations of the clock. In particular, we investigated the eﬀect of removing the positive feedback loop, previously identified as a significant factor in the determination of entrained phase. We found that while removal of positive feedback destroys selfsustained freerunning (DD) rhythms in the deterministic system, oscillations with significant amplitude persist when stochastic fluctuations are considered, demonstrating the greater robustness of the oscillatory mechanism in the stochastic model. In addition, we showed that knocking out the loop has little eﬀect on the stochastic variability of entrained phase for a given photoperiod, suggesting that positive feedback can be used to tune the phasephotoperiod relationship without introducing greater variation due to noise amplification. Finally, we considered sensitivity analysis techniques in order to identify the most influential parameters on the circadian function of the 12:12 LD system. We focused on the variations in FRQ expression resulting from perturbations to 5 putatively sensitive parameters, applying a local sensitivity method at diﬀerent time points within the 24hour cycle. By using a novel stochastic sensitivity measure based on histogram distance, we found that the FRQ waveform is maximally sensitive at the time it reaches is halfmaximum level, a molecular correlate for conidiation onset. We commented that this implicates stochasticity as a potential factor in the selection of this seemingly complex phase marker by evolution, rather than the phase of peak FRQ expression that is predicted to be maximally sensitive when variations in average FRQ level are considered. We conclude that while the local method we used only focuses around a specific point in the parameter space, it can still be informative, giving an idea about the impact of parameter changes on the behaviour of the system. In the future, we plan to apply some global methods in order to explore the full parameter space (or a meaningful subset of it) and to quantify the relationships between diﬀerent parameters.
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The use of stochastic simulation with our model merits some discussion, as it is characterised by some nonelementary reactions with complex kinetic laws, abstracting sets of interactions whose details are unknown. The use of Gillespie’s stochastic simulation algorithm (or its variants, such as GibsonBruck [31]) in the case of general kinetic laws has been discussed by several authors [36,37,38]. Rao and Arkin [36] showed that this approach is valid for some specific kinetic laws, such as MichaelisMenten and competitive inhibition. On the other hand, in [37] the authors demonstrated that this extension of Gillespie’s algorithm is not always appropriate. Here, we applied stochastic simulation paying particular attention to the interpretation of the simulation results and to their validation: in Sect. 5 we showed that the behaviour we obtain using our stochastic model is in agreement with the known behaviour of the system, and therefore we conclude that in this case the use of stochastic simulation is appropriate.
Acknowledgements The authors thank Jane Hillston and Andrew Millar for their helpful comments. The Centre for Systems Biology at Edinburgh is a Centre for Integrative Systems Biology (CISB) funded by BBSRC and EPSRC, reference BB/D019621/1. At the time of writing this paper, Federica Ciocchetta was a research fellow at the University of Edinburgh, supported by the U.K. Engineering and Physical Sciences Research Council (EPSRC) research grant EP/C543696/1 “Process Algebra Approaches to Collective Dynamics”. Andrea Degasperi is supported by a University of Glasgow Lord Kelvin/Adam Smith scholarship. Maria Luisa Guerriero is supported by EPSRC grant EP/E031439/1 “Stochastic Process Algebra for Biochemical Signalling Pathway Analysis”.
Supplementary Material The BioPEPA source files, the generated Dizzy files, some further information on the model and supplementary figure S1 are available online on the BioPEPA web page http://homepages.inf.ed.ac.uk/jeh/BioPEPA/ACDG_CMSB09_Supplement.zip .
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Quantitative Pathway Logic for Computational Biology Michele Baggi1, Demis Ballis2 , and Moreno Falaschi1 1
Dip. di Scienze Matematiche e Informatiche Pian dei Mantellini 44, 53100 Siena, Italy {baggi,moreno.falaschi}@unisi.it 2 Dip. Matematica e Informatica Via delle Scienze 206, 33100 Udine, Italy
[email protected] Abstract. This paper presents an extension of Pathway Logic, called Quantitative Pathway Logic (QPL), which allows one to reason about quantitative aspects of biological processes, such as element concentrations and reactions kinetics. Besides, it supports the modeling of inhibitors, that is, chemicals which may block a given reaction whenever their concentration exceeds a certain threshold. QPL models can be speciﬁed and directly simulated using rewriting logic or can be translated into Discrete Functional Petri Nets (DFPN) which are a subclass of Hybrid Functional Petri Nets in which only discrete transitions are allowed. Under some constraints over the anonymous variables appearing in the QPL models, the transformation between the two computational models is shown to preserve computations. By using the DFPN representation our models can be graphically visualized and simulated by means of well known tools (e.g. Cell Illustrator); moreover standard Petri net analyses (e.g. topological analysis, forward/backward reachability, etc.) may be performed on the net model. An executable framework for QPL and for the translation of QPL models into DFPNs has been implemented using the rewritingbased language Maude. We have tested this system on several examples.
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Introduction
Pathway Logic (PL) [20] is a symbolic approach to the modeling and analysis of biological systems which is based on rewriting logic [14]. Rewriting logic is a logical framework which allows one to easily formalize systems. Within this framework, the states of a system are represented as elements of an algebraic data type, speciﬁed by an equational theory, while its behavior is modeled via rewrite rules describing local transitions between states. The process of application of rewrite rules generates computations, which —in the case of biological systems— correspond to pathways.
This work has been partially supported by the Italian MUR under grant RBIN04M8S8, FIRB project, Internationalization 2004.
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A PL model is an executable rewriting logic speciﬁcation representing a biological process which consists of an equational part where biological molecules, their states and locations are declared. Such entities may be decorated by metadata such as entity synonyms and categories, protein families, etc. Such information can be retrieved by standard databases (e.g. HUGO [7], UniProt/SwissProt [22]). A collection of rewrite rules specifying individual reaction steps establishing how the system may evolve. In the case of signal transduction, rewrite rules represent processes such as activation, phosphorylation, complex formation, or translocation. As an example, consider the following PL rewrite rule labeled C.reloc of a given PL model. rl[C.reloc] {CLm  clm A} {CLi  cli B} {CLc  clc C} => {CLm  clm A} {CLi  cli B [C  reloc]} {CLc  clc}
The left handside of the rule represents a generic cell with three locations: the membrane (location tag CLm) containing element A, the inside of the membrane (location tag CLi) containing element B and the cytoplasm (location tag CLc) containing element C. The rule speciﬁes that when such a conﬁguration is detected, then C is relocated in CLi. Note that Pathway Logic provides a very simple way to add spatial information over the biological element describing a cell as a set of locations containing elements. Unfortunately, such a description is only qualitative, since the speciﬁcation is not equipped with quantitative information regarding the elements into play. PL models are speciﬁed using Maude [5], a rewritinglogicbased language. Maude speciﬁcations representing the PL models are executable, therefore they give a direct support to model simulation. Moreover, using Maude builtin search capabilities, PL models can be naturally queried in order to ﬁnd pathways of interest that may (may not) be activated. Finally, Maude speciﬁcations can be translated into equivalent Petri net (PN) representations by means of the Pathway Logic Assistant (PLA) [21]. The beneﬁt of such PN representations generated by PLA is twofold. On the one hand, it allows one to verify PL models reusing eﬃcient modelchecking techniques which have been already developed for PNs; on the other hand it provides a visual and interactive denotation of the model. Our contribution. Although Pathway Logic may be very useful to model biological processes and provides a simple way to express the systems dynamics, it has some important limits: (a) PL only supports qualitative modeling of the biological events of interests. As a matter of fact, it provides no explicit way to add quantitative information to the models such as element concentrations in cell locations, levels of production as well as consumption of elements occurring in a reaction, reaction thresholds, etc. (b) PL does not provide adequate capabilities to express inhibitory actions occurring in biological reactions, which are very common e.g. in regulatory networks. Basically, an element acts as an inhibitor in a reaction if its concentration over a given threshold prevents the reaction to take place. Instead, the inhibitor let the reaction take place whenever its concentration is under the considered threshold.
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In this paper we provide an extension of Pathway Logic called Quantitative Pathway Logic (QPL for short) with the aim of overcoming the mentioned PL limits and obtaining a more precise approach to modeling while keeping the possibility to compute with and analyze these complex systems. More speciﬁcally, QPL eﬃciently integrates quantitative data (such as element concentrations, reaction thresholds, production and consumption rates) into PL models. Besides, it allows one to model reaction inhibitors. To manage the diﬀerent aspects of biological systems, we equip QPL speciﬁcations with two equivalent computational models following and adapting the PL approach of [21]. This allows one to adopt diﬀerent representations with different expressive capabilities for handling the complexity of the systems under examination. On the one hand, QPL speciﬁcations can be directly formalized and executed by using the Maude rewriting logic formalism. In this way, both model simulation and model search can take advantage of quantitative information to yield more accurate results. On the other hand, QPL models can be translated into an extension of the classical Petri nets called Discrete Functional Petri Nets (DFPN), which are basically Hybrid Functional Petri Nets (HFPN) [12] in which only discrete transitions are authorized. By using such a representation our models can be graphically visualized, simulated, and analyzed by means of well known tools (e.g. Cell Illustrator [13]). Plan of the paper. The rest of the paper is organized as follows. In Section 2 we brieﬂy recall some necessary notions about rewriting logic and the Maude language. Section 3 presents the QPL rewriting logic formalism for modeling biological processes. Besides, we show how to specify, simulate, search and modelcheck QPL models using the Maude language. In Section 4, we provide an alternative representation of QPL models which is based on DFPNs. We also provide a transformation between the two representations that, under certain constraints, is shown to preserve the computations. Section 5 describes the prototypical implementation of our methodology. In Section 6 we discuss related work and we draw some conclusions.
2
Preliminaries
Rewriting logic [14] is a simple computational logic very well suited as a semantic framework within which many diﬀerent models of computation, systems and languages can be naturally modeled. A rewrite theory, that is, a theory in rewriting logic is a triple R = (Σ, E, R), where (Σ, E) is the equational theory modulo which we rewrite, and R is a set of (possibly conditional) labeled rules. Σ, called the signature, speciﬁes the operators and the type structure of R as usual, while E is a set of (possibly conditional) equations. Rules in R are of the form (label : t0 ⇒ t1 if c) where t0 , t1 are terms, the rule lefthand side (lhs) and the rule righthand side (rhs) respectively, and c is an optional boolean term representing the rule condition. Variables may appear in rules and equations and are denoted by lower letters, while operators are denoted by identiﬁers whose ﬁrst character is a capital letter. A context C, is a term with a single hole, denoted by [ ], used to indicate the location of a rewrite application. C[t] is the
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result of placing t in the hole of C. A substitution σ is a ﬁnite mapping from variables to terms, σ(t) is the result of applying σ to term t. Rewrite proceeds modulo the equational theory E and it is accomplished by performing pattern matching modulo the equational theory. More precisely, given an equational theory E, a term u and a term v, we say that u matches v modulo E (or that u Ematches v) via substitution σ if there exists a context C such that C[σ(u)] =E v, that is, C[σ(u)] and v are equal modulo the equational theory E. Given a term t and rule r = (label : t0 ⇒ t1 if c), we say that t rewrites to r t via r (in symbols t → t ), if there exists a substitution σ s.t. t0 Ematches t via σ, t = C[σ(t1 )] and σ(c) holds (i.e. it is equal to true modulo E). A rk r1 computation over R is a sequence of rewrites of the form R s0 → s1 . . . → sk , with r1 , . . . , rk ∈ R. Rewrite theories can be encoded in Maude [5], a highperformance reﬂective language supporting both equational and rewriting logic programming, which is particularly suitable for developing domainspeciﬁc applications. Maude speciﬁcations can be executed and veriﬁed using some powerful language builtin operators such as rew (which generates a computation starting from an initial term), search (which generates all the possible computations starting from a given initial term), and modelCheck (which supports model checking w.r.t. the Linear Temporal Logic (LTL) [6]).
3
Quantitative Pathway Logic
A QPL model is a rewrite theory Q = (Σ, E, R) which models biological systems. A QPL model is naturally divided in two parts: the equational part and the rules part. The former allows one to represent the cellular states, while the latter speciﬁes the system dynamics. The Equational Part. This part corresponds to equational theory (Σ, E) of the QPL model Q. It provides sorts and operators useful to model molecular components and more in general all the entities involved in a biological system. As in the standard PL framework, the main sorts for entities include Chemical, Protein, Complex, which are all subsorts of sort Thing specifying a generic entity. Cellular compartments are identiﬁed by sort Location which provides location names to each compartment, while Modification is a sort used to classify Posttransactional protein modiﬁcations, which are deﬁned by the operator [  ] (e.g. the term [EgfR  act] represents the epidermal growth factor Egf receptor in an activated state). Besides that, we provide a special sort QThing which is represented by the pair (Thing, R+ ), where R+ speciﬁes the sort for the nonnegative real numbers. The sort QThing is employed to manage entity concentrations (e.g. (Erk, 3.3) might model the fact that the concentration of the the MitogenActivated Protein Kinase Erk is 3.3 units). We call occurrence any term of sort QThing. A soup is a set of occurrences and cellular compartments which is identiﬁed by type Soup. Now, a cell state is represented by a term of the form [cellType  locs], where cellType speciﬁes the type of cell and locs represents the contents of a
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cell organized by cellular compartments (or locations). Each location is modeled by a term of the form { locName  comp }, where locName is a name identifying the location (e.g. CLm may represent the cell membrane location), and comp is a soup in that location. The Rules part. Given a QPL model Q = (Σ, E, R), rules part is speciﬁed via the set of rewrite rules R, which contains rewrite rules formalizing individual reaction steps. In the case of signal transduction, rewrite rules represent processes such as activation, phosphorylation, complex formation, or translocation. Basically, as in PL, QPL rewrite rules transform a cell state into another via pattern matching modulo an equational theory. Moreover, such a transformation in QPL can take advantage of the quantitative information associated with the entities into play. In this context, it is very easy to deﬁne promoters (entities enabling a reaction when their concentration is over a certain threshold), inhibitors (entities blocking a reaction when their concentration is over a certain threshold), tests (entities not consumed by a reaction), and reaction rates modeled via consumption and production functions. Let us see some examples. Example 1 (Promoters and tests). Consider a reaction modeled by the following rewrite rule. ex1 : {CLi  cli (A, a)}{CLm  clm ([BGDP], b)(D, d)} ⇒ {CLi  cli (A, a/2.0)(C, a/2.0 + b)} {CLm  clm (D, d)} if a >= 3.5.
The rule states that, if we detect a cell state in which (1) an entity A with concentration a is inside the cell membrane (location CLi); (2) entities [BGDP] (i.e. entity B bounded to a molecule of Guanosine diphosphate) and D with concentrations b and d are on the border of the cell membrane (location CLm); then, A promotes the reaction whenever its concentration is greater or equal than 3.5 units and a new cell state is generated in which Reactants A and [BGDP] are consumed: the former is consumed according to the consumption function (a/2.0) and the latter is completely consumed. Entity C is produced inside the membrane according to the production function a/2.0+b, while D is a test whose concentration is left unchanged (its presence is necessary for the reaction to take place but there is no need to consume it). Note that production functions may depend on concentrations of several reactants. Instead, we assume that the consumption function of a reactant A must depend only on the concentration of A. Inhibitory behaviors are easily modeled by means of conditional rules. For this purpose, we ﬁrst deﬁne the auxiliary function checkInhibitors(s, (i1, t1 ) . . . (in , tn )) which takes in input a soup s, and a list of occurrences (ij , tj ) and returns true if there does not exist any occurrence (ij , cj ) in s such that cj ≥ tj . Now, an inhibitor ij is an entity located in some compartment s which prevents a reaction to take place whenever its concentration cj is over a certain threshold tj . This amount to saying that a set of inhibitors (i1 , . . . , in ) can block a reaction iﬀ checkInhibitors(s, (i1, t1 ) . . . (in , tn )) = false. Therefore, we can model a
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reaction containing inhibitors i1 and i2 by a rewrite rule of the form t ⇒ t if checkInhibitors(s,(i1, t1 )(i2 , t2 )). Example 2 (Inhibitors). Consider a reaction modeled by the following rewrite rule. ex2 : {CLi  cli (A, a)} ⇒ {CLi  cli (C, a)} if checkInhibitors(cli, (B, 4.0))
The rule states that, when there exists a cell state in which an entity A with concentration a appears inside the cell membrane (location CLi), then the cell state is transformed by consuming completely the reactant A and producing a units of entity C provided that there is no inhibitor B with concentration greater than 4.0 inside the cell membrane (i.e. in the location CLi). Note that B is not consumed by the reaction. 3.1
Simulation and Analysis of QPL Models
Quantitative Pathway Logic models are rewrite theories, hence executable speciﬁcations, since they describe system states and provide rules specifying the way in which states may change. In other words, we can directly exploit the Maude rewrite engine to run our models. In particular Maude supports the forward simulation which is the ﬁrst kind of analysis that can be carried out given such an executable speciﬁcation. It consists in running the model from a given initial state for a ﬁxed number of steps or until a steady state has been reached. It is very useful for initial exploration of the transition graph but not suitable for understanding the dynamics of systems having inﬁnite behaviors. Maude is also equipped with forward search facilities, which allow one to perform a breathﬁrst search of all rewrite paths generated for a given initial state. If the speciﬁcation is ﬁnite, such a search will ﬁnd all possible outcomes from a given initial state. Moreover, the search can be constrained to ﬁnd only states satisfying a given property or until a ﬁxed number of rewrite steps. To execute both forward simulation and forward search we need to provide an initial state. Initial states (called dishes) are encoded in our rewriting framework by terms of the form PD(out cellstate), where cellstate represents a cell state and out speciﬁes a soup of ligands and other molecular components in the cells surroundings which may interact with the cell. Example 3. Consider the following dish PD({(Egf, 2.0)}[HMEC{CLoempty}{CLm(EgfR, 1.0)(PIP2, 3.0)} {CLi([Hras − GDP], 4.0)(Src, 5.0)} {CLc(Gab1, 5.0)(Grb2, 6.0)(Pi3k, 2.0)(Plcg, 7.0)(Sos1, 1.0)}]).
The dish above contains the cell state of a cell of type HMEC which is made up of three locations CLo, CLi and CLc. Each location contains a soup of occurrences. Besides, the ligand Egf occurs in the cell surroundings with concentration 2.0. Now, given an initial state and a QPL model, we can analyze the behavior of the system by means of Maude forward simulation and search capabilities. Let us see some examples.
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Example 4. Assuming that a QPL model for the signal transduction network of the epidermal growth factor receptor (EgfR) is given, we may run the model starting from an initial cell state by means of the Maude rewrite command (abbreviated rew), that is, we may explore the behavior of the speciﬁed system for diﬀerent initial cell states. For instance, the Maude query rew [100] PD((Egf,2.0) [HMEC  {CLm  (EgfR,1.0) (PIP2,3.0)} {CLi  ([HrasGDP],4.0) (Src,5.0)}])
asks Maude to perform at most 100 rule applications (i.e. rewrite steps) to rewrite the given initial state and returns the ﬁnal cell state we reached. It is also possible to rewrite without specifying an upper bound on the number of rule applications. Since the model may be nondeterministic (i.e. there might be several computations starting from the same initial state), Maude selects only one of such computations by means of a predetermined rewrite strategy. Therefore, the returned cell state may represent only one of the possible system behaviors. Maude also provides the frew which allows one to implement userdeﬁned rewrite strategies. As explained in Example 4, the forward simulation explores only one possible model behavior. To analyze all possible model dynamics we may employ the Maude search feature as shown in the following example. Example 5. Assuming the same QPL model of the previous example, we want to know whether —starting from a dish containing a given concentration of the ligand Egf— it is possible to produce the MitogenActivated Protein Kinase Erk (as described in [20]) with a concentration greater than 3.6 units. Such analysis can be modeled by means of the following Maude forward search query: search [1] PD((Egf,2.0) [HMEC  {CLo  empty} {CLm  (EgfR,1.0)(PIP2,3.0)} {CLi  ([HrasGDP],4.0)(Src,5.0)} {CLc  (Gab1,5.0)(Grb2,6.0)(Pi3k,2.0)(Plcg,7.0)(Sos1,1.0)}]) =>+ PD(out:Soup [HMECcyto:Soup {CLi(Erk,k)}]) such that k > 3.6 .
The term before the right arrow denotes the initial state, while the one after the arrow speciﬁes the pattern of the state we are looking for along with the condition that Erk has to appear with a concentration higher than 3.6. Finally, also Maude Model checking [6] can be employed to analyze QPL models. Model checking enlarges the set of properties which can be investigated. While search only concerns with properties of individual states, model checking deals with properties of computations (i.e. pathways). In this context, the modelchecker is typically asked to check the assertion that there is no computation starting from the given initial state satisfying the property of interest; thus a path can be extracted from a counterexample, if one is found. Example 6. By using Maude modelCheck function we can easily verify whether, starting from a given initial state istate containing Egf, it is possible to activate the entity Src with a concentration greater than 4.5 units without having previously produced the entity RalaGDP. To this purpose, we deﬁne a parametric
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property entAct(e,n), which is satisﬁed when the CLi location contains an occurrence of entity e with a concentration higher than n. The property might be speciﬁed as follows eq PD(out:Soup [HMECcyto:Soup {CLicli:Soup(e,k)}]) = entAct(e,n) = k > n . Now, we can model the desired analysis by means of the following Maude modelchecking query: red modelCheck(istate, []∼( entAct([Srcact],4.5) /\ (entAct([Srcact],4.5) > entAct(RalaGDP,0.0))))
The query is expressed in linear temporal logic and consists of a conjunction of two subqueries, the former asks for the activation of Src with a concentration of 4.5 units and the latter asks for a sequence of events (expressed by the > operator) where the production of Rala strictly follows the activation of Src.
4
Representing QPL Models via DFPNs
QPL models can be represented by means of Discrete Functional Petri Nets (DFPN) which are restricted HFPNs[12] able to model quantitative aspects of a given system by means of functional discrete transitions. The advantage of such an alternative formalism is twofold: on the one hand, graphical representations are naturally derived from DFPNs, which allow one to visualize the model of interest and to graphically interact with it using common tools available on the market (e.g. Cell Illustrator); on the other hand, analysis of DFPNs can proﬁt from wellknown techniques which have been already developed for the Petri net settings. In what follows, we formalize DFPNs by borrowing terminology and notation from [12]. 4.1
Discrete Functional Petri Nets
Definition 1 (Marking). Given a ﬁnite set of places P , a marking of P is a mapping M : P → R+ . Given p ∈ P , M (p) is called the mark of p. We denote the set of all possible markings of P by M. Given two markings M and M of P , we say that M ≥ M iﬀ for each p ∈ P , M (p) ≥ M (p). Moreover, we say that M and M are incomparable if M M and M M . Let M be the set of all markings of the set of places P , we denote the set of all functions mapping a marking of P into a nonnegative real number as FP = {f  f : M → R+ }. Functions in FP are called update functions. Definition 2 (Discrete Functional Petri Nets). A Discrete Functional Petri Net (DFPN) is a triple P = (P, T, C) where P = {p1 , . . . , pn } is a nonempty ﬁnite set of places, T = {t1 , . . . , tm } is a nonempty ﬁnite set of transitions such that P ∩ T = ∅, and C is a tuple (P T, T P, a, w, u) deﬁned as follows: – P T ⊆ P × T and T P ⊆ T × P . Elements in P T (resp. T P ) are called input connectors (resp. output connectors). Each connector has a connector type which is given by a function a : P T ∪ T P → {process, test, inhibitor, output}. Input connectors whose type is process (resp. inhibitor, test) are also called process (resp. inhibitor, test) connectors.
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– w : P T → R+ is a mapping called threshold labeling assigning nonnegative real numbers to input connectors. – u : P T ∪ T P → FP is a partial mapping called update labeling such that u(c) is deﬁned iﬀ a(c) ∈ {process, output} (i.e. mapping u assigns update functions to process and output connectors only). Connectors in a DFPN are labeled by threshold and update labeling. More specifically, threshold labeling puts nonnegative real numbers (i.e. thresholds) on input connectors (p, t) and are used to ﬁx the minimum threshold on the mark of place p which is required to enable/disable transition t. Update labeling decorates both process and output connectors with update functions and is employed to change the marks of places involved with transitions ﬁring. The enablement relation of a transition t in a DPFN depends on the type of the input connectors (p, t). Basically, an inhibitor connector enables transition t when the mark of p is under a certain threshold; process and test connectors enable transition t whenever the mark of p is over the threshold. More formally, Definition 3. Let P = (P, T, C), where C = (P T, T P, a, w, u), be a DFPN. Given a transition t ∈ T and a marking M ∈ M of P , we say that t is enabled in M iﬀ for each input connector c = (p, t) ∈ P T the following conditions hold: (1) M (p) < w(c) if a(c) = inhibitor (2) M (p) ≥ w(c) if a(c) = inhibitor. Otherwise transition t is said to be disabled in M . We denote the set of all the transitions which are enabled in M by E(M ). Now, given a DFPN P, we can deﬁne computations over P as follows. Definition 4. Let P = (P, T, C), where C = (P T, T P, a, w, u), be a DFPN. Let M, M ∈ M be two markings of P and t be a transition in T such that t t ∈ E(M ). Then, M evolves into M using t in P (in symbols, P M → M ) iﬀ M = DFPN One Step(P, M, t), where DFPN One Step(P, M, t) is a function deﬁned as follows: DFPN One Step(P, M, t) for each (p, t) ∈ P T with a(p, t) = process: M (p) ← M (p) − u(p, t)(M ) for each (t, p) ∈ T P : M (p) ← M (p) + u(p, t)(M ) return M t
t
t
A computation in P is a (possibly inﬁnite) sequence P M0 → 0 M1 → 1 M2 → 2 . . . Marking M0 is called initial marking. Roughly speaking, a marking M can evolve into a marking M by ﬁring an enabled transition t producing M from M in the following way: for each process connector (p, t) the mark M (p) is consumed by applying the update function labeling (p, t); for each output connector (t, p), the mark M (p) is increased by applying the update function labeling (t, p); for each inhibitor/test connector (p, t), the mark M (p) is left unchanged Transitive ( →+ ) and transitive and reﬂexive ( →∗ ) closures of relation → are deﬁned in the usual way. Note that several transitions can be enabled at the same time in a DFPN producing nondeterministic computations.
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Translating QPL Models into DFPNs
Given a QPL model expressed by a rewrite theory Q = (Σ, E, R), we can easily derive a DFPN having the same model behavior which will be denoted by PQ . Basically, the translation procedure produces a DFPN transition for each rule r belonging to R. Let us see how the translation of a single rule into a transition proceeds. Let r = (l : t ⇒ t if c), we deﬁne OL(r) = {(e, q, loc)  occurrence (e, q) appears in t in location loc}, OC(r) = {(e, q, loc)  entity e appears in predicate checkInhibitors in c associated to location loc}, and OR(r) = {(e, q, loc)  occurrence (e, q) appears in t in the location loc}1 . First of all, for each (e, q, loc) ∈ OL(r) ∪ OR(r) ∪ OC(r), we deﬁne a place e, loc whose marking M (e, loc) is represented by q. Intuitively, each place in the resulting DFPN will model a given entity e appearing in a compartment loc, while the concentration q provides information regarding the mark of the place. Then, we generate the transition l, where l is the label of the rule r under examination. Transition l is connected to the generated places via input/output connectors in the following way. For each (e, q, loc) ∈ OL(r) we generate an input connector con = (e, loc, l) whose type a(con) depends on the role of the entity e in the original rule r (i.e. is e a process or a test?). For each (e, loc) ∈ OC(r) we generate an input connector con = (e, loc, l) whose type a(con) = inhibitor. For each (e, q, loc) ∈ OR(r) if a(e, loc, l) = test we generate an output connector con = (l, e, loc) whose type a(con) = output. Finally, thresholds and update functions are deﬁned according to the expressions appearing in the rule condition. Note that rule’s variables not representing concentrations are not encoded into the net PQ . Such variables have only an auxiliary purpose in the QPL model. Indeed, they are used to enable pattern matching, which is an evaluation mechanism not employed in DFPNs, and hence they do not have a counterpart in the resulting net model2 . For further details, the complete description of the translation method can be found in [3]. Figure 1 shows DFPN obtained by the the translation of rules ex1 and ex2 of Examples 1 and 2. A PL dish, that is an initial state for a QPL model, is naturally encoded into an initial marking for the resulting DFPN in which we assign a given mark to (some of) the places e, loc of the net. 4.3
Model Equivalence
Given a QPL model Q, the resulting DFPN PQ is equivalent to Q in the sense that computations are preserved. In other words, any computation over Q is mapped to a computation over PQ and vice versa. The equivalence holds under certain conditions we need to enforce on the translation. In the remainder of 1 2
As usual locations are identiﬁed by their location name. Due to lack of space and for the sake of readability, we have not treated the case of entity variables which can range over a ﬁnite set of values (e.g. a variable representing a class of ligands which may interact with a given receptor). A full explanation of such a case can be found in the technical report [3].
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H = (P, T, C) P = {, , , , , , } T = {ex1, ex2} C = {In, Out, a, w, u} In = {(,ex1), (,ex1), (,ex1), (,ex2), (,ex2)} Out ={(ex1,), (ex2,)} ⎧ process if c ∈ {(,ex1), (,ex1), (,ex2)} ⎪ ⎪ ⎨ test if c =(,ex1) a(c) = inhibitor if c =(,ex2) ⎪ ⎪ ⎩ output if c ∈ {(ex1,), (<ex2,C,CLi>)} 3.5 if c =(,ex1) w(c) = 4.0 if c =(,ex2) ⎧ A CLo q/2.0 if c =(,ex1) ⎪ ⎪ ⎪ ⎪ if c =(,ex1) ⎨ [BGDP] CLm q u(c) = A CLo q/2.0 + B CLm q if c =(ex1,) ⎪ ⎪ A CLi q if c =(,ex2) ⎪ ⎪ ⎩ A CLi q if c =(ex2,) Fig. 1. DFPN encoding of rules ex1 and ex2
this section, we provide a brief explanation of this result. We refer the reader to the technical report [3] for more details and the complete proofs. Our approach follows and adapts the approach presented in [21] by Talcott and Dill for establishing the equivalence between Pathway Logic and standard Petri nets. Let Q be a QPL model and PQ the DFPN obtained from Q. Let SQ be the set of all possible (ground) cell states of Q and MQ be the set of all possible markings of PQ . We deﬁne a mapping s2m : SQ → MQ which maps a cell state s ∈ SQ into a marking Ms ∈ MQ such that, for each entity e appearing in a location loc with concentration q, Ms (e, loc) = q. We deﬁne the inverse mapping of s2m by m2s : MQ → SQ , and thus we have m2s(s2m(s)) = s for any cell state s ∈ SQ . Since cell states are terms which can be built upon contexts and holes, we extend s2m to contexts and holes in such a way that s2m(C[t]) = s2m(t) ∪ s2m(C). Therefore, given a place e, loc of PQ , which is obtained from a cell state s = C[t] ∈ SQ , where C is a context with a hole at location locC , the mark of e, loc w.r.t. the mapping s2m(C[t]) is deﬁned as s2m(C[t])(e, loc) =
s2m(t)(e, loc) s2m(C)(e, loc)
if loc = locC otherwise
In order to guarantee the computational equivalence between QPL models and their DFPN counterparts, we need to enforce a constraint over the rules of QPL models. Let us see an example illustrating some issues related to the model translation which may arise.
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Example 7. Consider the two following QPL rules: r1 : {CLm  (A, a) {CLc  cyto (B, b)}} ⇒ {CLm  (A, a) ([B − act], b) {CLc  cyto}} r2 : {CLm  clm (A, a) {CLc  cyto (B, b)}} ⇒ {CLm  clm (A, a) ([B − act], b) {CLc  cyto}}
where cyto and clm are “anonymous” variables matching any other component located in the cytoplasm or cell membrane respectively. Consider now the state s = {CLm(A, 3.3)(C, 4.7){CLc(B, 0.1)(D, 0.9)}}. The rule r2 applies to s but r1 does not because it lacks variable clm which enables the pattern matching between s and the lhs of r1 on the location Clm. On the other hand, r1 and r2 are translated into transitions t1 and t2 which are equal modulo renaming of the transition labels. In particular, t1 and t2 connect the same places via the same input/output connectors. Therefore, t1 is enabled whenever t2 is, and vice versa. Clearly, this fact generates a discrepancy between the computational behaviors of the QPL model and the corresponding DFPN. To avoid the situation presented in Example 7, we basically forbid rules like r1 from being speciﬁed in QPL models. Definition 5. Let Q = (Σ, E, R) be a QPL model. Then Q is wellspeciﬁed iﬀ for each rewrite rule r ∈ R, each location Loc appearing in the lhs or rhs of r contains a variable loc. Notice that the QPL model of Example 7 is not wellspeciﬁed. The equivalence result between the two computational models is stated by the following theorem. Theorem 1. Let Q = (Σ, E, R) be a wellspeciﬁed QPL model and PQ be the DFPN obtained from Q. Let r2t be a function mapping each rule r ∈ R into the corresponding transition r2t(r) of PQ . Then, the following result hold: r
r
r2t(r1 )
r2t(rk )
1 k Q s0 → s1 . . . → sk ⇔ PQ s2m(s0 ) −→ s2m(s1 ) . . . −→ s2m(sk )
where si ∈ SQ , rj ∈ R
5
Implementation
We implemented the framework presented so far in a prototypical system written in Maude, which is publicly available along with some examples at http://users.dimi.uniud.it/~michele.baggi/qpl. Basically, our system extends the Pathway Logic data structures and mechanisms to cope with quantitative information. Our system allows one to specify and analyze Quantitative Pathway Logic models by means of Maude language features. In particular, it is possible to exploit Maude builtin operators to easily express queries to simulate, search and modelcheck the models under examination. We tested the prototype on several small/medium size biological systems (e.g. the circadian rhythm in Drosophila) achieving rather promising results. In the future, we plan to provide a thorough experimental evaluation on realsize case studies assisted
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Fig. 2. Cell Illustrator screenshot of the EgfR pathway model
by the biologists of our group. The prototype is also equipped with a model translator which allows one to automatically derive the corresponding DFPN from the given QPL model. On the net representation, we can apply wellknown Petri net analysis methodologies such as topological analysis for relevant subnet detection, backward/forward reachability analysis, etc. For more information about DFPN analyses, please refer to our technical report [3]. Moreover, we provide the possibility to export DFPN descriptions in the Cell System Markup Language (CSML) [17], an XML format for modeling biopathways which covers widely used data formats, e.g. CellML[11], SBML[8]. Then, CSML representations of DFPNs can be imported in Cell Illustrator [13], which is a software tool by means of which we can visualize and graphically interact with the DFPN models. In Figure 2 a screenshot of the Cell Illustrator application with a small fragment of the EgfR pathway model is shown. Here, discrete transitions are identiﬁed by solid rectangles, while continuous places are represented by double circles. Dashed arrows and solid arrows stand for process connectors and test connectors, respectively; ﬁnally, labels identify thresholds and update functions.
6
Related Work and Conclusions
There are many diﬀerent computational models of biological processes, depending on the aspects we want to focus on. Our approach to biopathway modeling basically exploits rewrite theories [14] and Petri nets [16] with the aim of taking advantage of the beneﬁts conveyed by both these formalisms. The Petri net formalism allows one to model networks of reactions describing processes as well as process execution with a graphical representation that naturally corresponds to conventional representation of biochemical networks. Standard Petri nets have a limited expressiveness (e.g. they cannot describe quantitative aspects of the model or capture reaction inhibition eﬀects). To this respect, many powerful variants of the Petri net formalism have been developed along with a plethora of techniques and tools for system speciﬁcation and analysis (e.g. see [4,9,10]).
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Nonetheless the speciﬁcation of large biological systems involving a huge number of states and transitions might result in a rather complex task, since Petri nets make system state and state changes explicit through marking of places and transitions which are required to be individually speciﬁed. On the other hand, rulebased formalisms directly model states of molecular components and state changes by means of rules in which rule patterns may subsume several distinct states. This implies that rulebased speciﬁcations are in general more concise and easy to write than Petri nets. As already explained, Pathway Logic (PL) [20] represents biological processes by means of rewriting logic theories. PL allows one to only model the qualitative aspects of the processes. A recent extension of the Pathway Logic [1] has been proposed to represent and reason about semiquantitative and probabilistic aspects of biological processes. Basically, this approach annotates reaction rules with aﬃnity information that can be used to implement distinct simulation strategies which can also include timing information. Although this approach improves expressiveness of standard PL, it only handles a semiquantitative modeling of biological processes which does not allow us to deﬁne complex reaction kinetics. In contrast, QPL fully supports quantitative information such as element concentrations, reaction enabling thresholds, production and consumption rates in a concise and quite intuitive way. Besides, from the QPL model, we can automatically generate a graphical DFPN representation which can be displayed and intuitively manipulated via Cell Illustrator. The extension we propose is the ﬁrst step towards the development of a general PL formalism for the full speciﬁcation of hybrid models (specifying both discrete and continuous components) and their stochastic counterparts. As future work, we intend to (i) add probabilistic behaviors to our models in the style of [1], and (ii) endow our speciﬁcation language with timing information following the approach of [18], where hybrid systems are formalized by means of suitable rewrite theories.
References 1. Abate, A., Bai, Y., Sznajder, N., Talcott, C., Tiwari, A.: Quantitative and Probabilistic Modeling in Pathway Logic. In: 7th IEEE International Conference on Bioinformatics and BioEngineering, pp. 922–929. IEEE Xplore, Los Alamitos (2007) ˇ ans, K., Jonsson, B., Tsay, Y.: Algorithmic analysis of programs 2. Abdulla, P.A., Cer¯ with well quasiordered domains. Information and Computation 160(12), 109–127 (2000) 3. Baggi, M., Ballis, D., Falaschi, M.: Applications to Systems Biology of Quantitative Pathway Logic. Technical report (2009), http://users.dimi.uniud.it/~ michele.baggi/qpl/BBF09tr.pdf 4. Chaouiya, C., Remy, E., Thieﬀry, D.: Petri net modelling of biological regulatory networks. Journal of Discrete Algorithms 6(2), 165–177 (2008) 5. Clavel, M., Dur´ an, F., Eker, S., Lincoln, P., Mart´ıOliet, N., Meseguer, J., Talcott, C.: The Maude 2.0 System. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 76–87. Springer, Heidelberg (2003)
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6. Eker, S., Meseguer, J., Sridharanarayanan, A.: The maude LTL model checker and its implementation. In: Ball, T., Rajamani, S.K. (eds.) SPIN 2003. LNCS, vol. 2648, pp. 230–234. Springer, Heidelberg (2003) 7. Eyre, T.A., et al.: The HUGO Gene Nomenclature Database, 2006 updates. Nucleic Acids Research 34(Database issue), 319–321 (2006) 8. Gauges, R., Rost, U., Sahle, S., Wegner, K.: A model diagram layout extension for SBML. Bioinformatics 22(15), 1879–1885 (2006) 9. Genrich, H.J., K¨ uﬀne, R., Voss, K.: Executable Petri net models for the analysis of metabolic pathways. International Journal on Software Tools for Technology Transfer 3(4), 394–404 (2001) 10. Gilbert, D., Heiner, M., Lehrack, S.: A Unifying Framework for Modelling and Analysing Biochemical Pathways Using Petri Nets. In: Calder, M., Gilmore, S. (eds.) CMSB 2007. LNCS (LNBI), vol. 4695, pp. 200–216. Springer, Heidelberg (2007) 11. Lloyd, C.M., Lawson, J.R., Hunter, P.J., Nielsen, P.F.: The CellML Model Repository. Bioinformatics 24(18), 1367–2123 (2008) 12. Nagasaki, M., Doi, A., Matsuno, H., Miyano, S.: A versatile petri net based architecture for modeling and simulation of complex biological processes. Genome Informatics 15(1), 180–197 (2004) 13. Nagasaki, M., Doi, A., Matsuno, H., Miyano, S.: Genomic Object Net: A platform for modelling and simulating biopathways. Applied Bioinformatics 2, 181–184 (2004) 14. Mart´ıOliet, N., Meseguer, J.: Rewriting Logic: Roadmap and Bibliography. Theoretical Computer Science 285(2), 121–154 (2002) 15. Minsky, M.L.: Computation: ﬁnite and inﬁnite machines. PrenticeHall, Inc., Upper Saddle River (1967) 16. Murata, T.: Petri Nets: Properties, analysis and applications. Proc. IEEE 77(4), 541–580 (1989) 17. Nagasaki, M., Saito, A., Li, C., Jeong, E., Miyano, S.: Systematic reconstruction of TRANSPATH data into Cell System Markup Language. BMC Systems Biology 2(1) (2008) ¨ 18. Olveczky, P.C., Meseguer, J.: Speciﬁcation of realtime and hybrid systems in rewriting logic. Theoretical Computer Science 285(2), 359–405 (2002) 19. Reinhardt, K.: Reachability in Petri Nets with Inhibitor Arcs. Electronic Notes in Theoretical Computer Science 223, 239–264 (2008) 20. Talcott, C.: Pathway logic. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 21–53. Springer, Heidelberg (2008) 21. Talcott, C., Dill, D.L.: Multiple representations of biological processes. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 221–245. Springer, Heidelberg (2006) 22. The uniprot consortium. The Universal Protein Resource (UniProt). Nucleic Acids Research 35(Database issue) (2007)
A PrizeCollecting Steiner Tree Approach for Transduction Network Inference Marc BaillyBechet1,2, Alfredo Braunstein1 , and Riccardo Zecchina1 1
Microsoft TCI Research, Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 2 Universit´e de Lyon, F69000, Lyon, CNRS, UMR5558, Laboratoire de Biom´etrie et Biologie Evolutive, F69622, Villeurbanne, France
[email protected],
[email protected],
[email protected] Abstract. Into the cell, information from the environment is mainly propagated via signaling pathways which form a transduction network. Here we propose a new algorithm to infer transduction networks from heterogeneous data, using both the protein interaction network and expression datasets. We formulate the inference problem as an optimization task, and develop a messagepassing, probabilistic and distributed formalism to solve it. We apply our algorithm to the pheromone response in the baker’s yeast S. cerevisiae. We are able to ﬁnd the backbone of the known structure of the MAPK cascade of pheromone response, validating our algorithm. More importantly, we make biological predictions about some proteins whose role could be at the interface between pheromone response and other cellular functions.
1
Introduction
Living cells need to react to a wide spectrum of changes –physical, chemical or biological– in their environment [1]. Conversely the cell reactions span from the activation of smallscale processes, e.g. synthesis of precise molecular components or excretion of others, to complex changes in the global cellular state, such as the diauxic shift or pheromone response and mating [2] in yeast. In order for the cell to survive, these changes must be tightly regulated. One type of regulation occurs through signaling cascades, which represents how the information propagates inside a cell, from receptor proteins to transcription factors and other eﬀector proteins. At the molecular level, this information transits by activation or inactivation of speciﬁc signaling proteins. Activation mechanisms include a variety of proteinprotein interactions such as conformation changes, or dimerization [3]; one of the most studied is the wellknown phosphorylationdephosphorylation system provided by kinases [4]. Known signaling cascades show desirable properties, from a system point of view: they act as lowpass ﬁlters, ensuring an adequate cell response only when external stimuli are above the molecular noise level [5], but they also provide signal ampliﬁcation [6]. Recently, it was also shown that information could travel in both directions on signaling P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 83–95, 2009. c SpringerVerlag Berlin Heidelberg 2009
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cascades, due to chemical equilibrium shifts in the cascades[7]. These properties can be used by the cell to tune the signal propagation and therefore the response to the environment. The intersection of the signaling pathways forms the transduction network, whose nodes are proteins and whose edges represent protein interactions transmitting information. Due to the many interconnections between diﬀerent signaling cascades in the transduction network, a precise regulation of the crosstalk between diﬀerent pathways is necessary. One way to ensure pathway speciﬁcity in answer to a given signal is the usage of scaﬀold proteins which will speciﬁcally bind to other members of a given signaling cascade, increasing speciﬁcity of the response [8,9]. On the other hand, one way to diﬀuse signal to many pathways is to root them all to the same activator protein. The complexity of these crossinteractions has allowed evolution to shape these pathways so as to be very eﬃcient in sensing and adjusting to the environment, but makes them very diﬃcult to study independently and even to identify precisely. In this work we tackle the issue of transduction network inference from proteomics and transcriptomics data. Phosphoproteomics works have been led to reconstruct these cascades, but are still very expensive and time consuming. At the algorithmic level, this problem has been widely studied, mainly by inference of linear cascades. In this context Scott et al. [10] developed an algorithm based on colorcoding to infer linear subparts of the transduction network, and found with good accuracy the known MAPK kinases cascades. White et al. [11] made a step forward by looking for transduction networks as a superposition of shortest paths on the protein interaction network (PIN). The focus of their method is to unbias the solution tree from the high connectivity bias, as often hubs of the PIN tend to be overrepresented in the inferred networks, as a consequence of their high inbetweenness. Other works about the inference of transduction network include [12], who introduced a Steiner tree formalism to recover this network based on expression data and an existing PIN. This formalism states that the transduction network is a subtree of the global protein interaction network which contains all proteins of a given subset, named terminals, deﬁned by the user. This subset is composed of proteins known to be part of the signaling network, or selected via another criterion such as expression level. The problem is then to reconstruct such a tree, respecting also other combinatorial constraints such as e.g small tree size or ﬁxed tree depth. This last approach was recently developed by [13], who used an integer linear programming relaxation to ﬁnd subnetworks involved in signal transduction, improving the algorithmic performance. The previously cited approaches have been eﬀective at ﬁnding already known signaling cascades, but made few predictions, mainly because of the time constraints of the available techniques in the ﬁeld of combinatorial inference problems. Indeed the Steiner tree problem [14] is NPhard and classical algorithms allowing to solve it at the probabilistic level are slow. Here we provide a new method from the class of messagepassing algorithms to infer a Steiner tree from a weighted graph, which directly applies to infer transduction networks from a PIN and expression data. From a numerical point of view, messagepassing algorithms are
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probabilistic and distributed, allowing for a very fast resolution of inference problems [15], even for large networks. Moreover, our algorithm does not need a priori selected terminals (i.e proteins of interest), and compute the transduction network as a whole, instead of a sum of linear subparts, as was done in previous works. This results in a high exploratory power of combinatorial eﬀects that could uncover biologically meaningful crosstalk. We apply it to the pheromone response in S. cerevisiae; results show that we are able to reconstruct accurately known pathways, to infer how the signal propagates in other signaling cascades of the cell, and to make functional predictions about a new group of genes implied in the pheromone response.
2
Material and Methods
The rationale of our model is that the transduction network is a subtree of the PIN, which should be composed with links of the PIN corresponding to real protein interactions, and proteins being of biological relevance for the biological process under study. Indeed, proteinprotein interactions detected in proteomic assays contain a high fraction of false positives [16], creating the need to take into account in our model the statistical conﬁdence we have for each link of the PIN. As proteomics data are still scarce, whether expression data are nowadays available in huge quantities, we hypothesized, as was previously done by [13,17,18,10,12], that genes being diﬀerentially expressed during the activation of the signaling pathway encode proteins being necessary for the signaling response itself, and employed expression data to measure the relative importance of each protein in a given environmental context. Therefore we could model the transduction network inference as an optimization problem, given weights for every edge of the PIN, to represent the propensity of the edge to be a false positive, and prizes for the nodes, proportional to the level of diﬀerential expression of the corresponding genes in the expression data relative to the phenomenon under study. 2.1
Inference of the Transduction Network
In general terms, we are interested in ﬁnding a “minimal” subnetwork that is connected to a given protein node, known as the root. We will model this problem as a PrizeCollecting Steiner Tree on Graphs problem (see e.g. [19,20]). Given a network G = (V, E) with positive (real) weights {wl : l ∈ E} on edges and {wn : n ∈ V } on vertices, we are interested in ﬁnding the connected subnetwork that minimizes the following quantity: C= wl − λ wn (1) links
nodes
It is easy to see that such network must be a tree (links closing cycles can be removed, lowering C). λ is a parameter regulating the balance between optimization of the two terms of the sum.
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This problem is known to be NPHard, implying that is unlikely that an algorithm that can eﬃciently solve any instance of the problem exists. To solve it we will use a small variation of an extremely eﬃcient heuristics based on belief propagation developed on [14] that is known to be exact on many classes of random networks [14,21]. The algorithm iterates the following set of equations for the quantities {ψij }(ij)∈E (called ”messages”) to a ﬁxed point: t+1 t ψji (dj , pj ) = −cjpj + max ψkj (dk , pk ) (2) (kj)∈E\(ij)
f (dk ,pk ,dj ,pj ) =0
where di ∈ D = {0, ..., D −1}, pi ∈ V (i)∪{∅}, ψji : D ×V (i)∪{∅} → R and fij is a characteristic function that ensures the condition pi = j ⇒ pj = ∅, dj = di − 1 deﬁned as follows: fij = gij gji gij = (1 − δpj ,i (1 − δdi ,dj−1 ))(1 − δpj ,i δpi ,∅ ) On a ﬁxed point, the following quantities (”ﬁeld”) are computed: t ψj (dj , pj ) = max ψkj (dk , pk ) (kj)∈E
f (dk ,pk ,dj ,pj ) =0
Then a tree T ∗ is built from the parenthood relations deﬁned as follows: deﬁne d∗j , p∗j = arg maxdj ,pj ψj (dj , pj ). Then if p∗j = ∅, deﬁne the parent of j as pj . Otherwise, j does not belong to T ∗ . With a minimal nondegeneracy assumption on the initial ﬁelds, it is relatively straightforward to verify that with variables d∗j , p∗j , fij = 1 ∀(ij) ∈ E and this implies that T ∗ is indeed a tree. It can be proved in some limit cases that the algorithm is optimal, and veriﬁed experimentally that it generally gives an excellent approximation to the optimal. For more details see [21]. 2.2
Data Source and Definition of the Weights
The yeast protein interaction network (PIN) was built by combining data from two databases : DIP [22] and MIPS [23]. The combined network has 5217 nodes and 22637 edges. To deﬁne their weights, edges were divided in two categories: a high conﬁdence one, containing links extracted from smallscale experiments or found many times; and a low conﬁdence one, containing links found only once in a largescale experiment. We deﬁned the two corresponding weights so as to maximize the correlation of our weight set and the one of [24], giving a weight wl = 1 for high conﬁdence edges (24.9% of the PIN) and a weight wl = 1.74 for low conﬁdence edges. The choice of this weight set as a reference is based on the observation that it is one of the most reliable [25], and does not derive the weights from expression data. We analyzed 56 expression datasets from [26]. We computed node prizes for each dataset in a classical way by taking wn = − log(pn ), where pn is the pvalue of diﬀerential expression of node n in the corresponding microarray. Though, a high prize was attributed to genes having a signiﬁcant pvalue in the expression data.
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The 56 datasets were analyzed independently with values of λ ranging from 0.05 to 0.9. The chosen root was the receptor protein STE2 in datasets comparing cells submitted or not to pheromone α action. In datasets with an artiﬁcially overexpressed gene under GAL4 promoter control, this gene was chosen as the root. If the strain used contained deletions, the corresponding genes were removed from the PIN prior to inference. In datasets comparing deleted strains to wild type strains without exposition to pheromone, deleted genes were selected as roots. 2.3
Statistical Analyses
Functional homogeneity of the trees inferred for each expression microarray and each value of λ was assessed by comparing the number of GO Slim annotations [27] shared by interacting proteins in the inferred Steiner trees, and random trees with same root and size, with edges probabilistically weighted as in the real data or not. Random trees were generated 50 times and results were averaged. Steiner proteins were deﬁned as proteins present in the Steiner tree with wn < λ1 : such proteins have a local cost to be added in the tree, which has to be compensated. Enrichment of the inferred trees in proteins of interest was estimated by comparison with random trees generated with permuted expression data, for λ = 0.2 (30 iterations).
3
Results
Our algorithm infers an organism transduction network, using as a support the PIN and expression data to ﬁnd a Steiner tree maximizing the level of diﬀerential expression of its nodes (genes) and built preferentially with edges of high conﬁdence. The free parameter λ (see Mat. Meth.) regulates the balance between optimization on the edges and on the nodes, and therefore regulates the tree size. For each microarray given as input, the Steiner tree found is a representation of the transduction network activated in the corresponding condition. The Steiner trees representing transduction networks were inferred in 56 expression datasets from a study about pheromone response [26], with 7 diﬀerent values of λ. A statistical description of the trees found is provided in Table 1. As expected, both the frequency of highcost links selected and the average tree size increase with λ. As an integrity check we analyzed the correlation between the tree size and the average prize wn of the nodes in the datasets, which is a direct measure of the numbers of genes diﬀerentially expressed on the microarray (Fig 1). As expected, the average tree size increases both with λ and average node prize; indeed, this second dependence even seems linear, a property that could be interesting to detect anomalies in the inferred trees. An averaged representation of the trees found for λ = 0.2 is given in Fig 2. Proteins usually found as members of the pheromone response pathway are present, such as FUS3, GPA1 or SST2; some missing intermediates appear for higher
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Table 1. Statistical properties of the trees inferred. One can see the evolution of the average tree properties with increasing values of the parameter λ, notably the increase in average tree size and decrease in the fraction of Steiner proteins found. The global fraction of highcost links in the PIN is 75.1%, notably higher than the fraction present in the inferred trees. λ
Tree size (# prot)
0.05 0.1 0.2 0.3 0.5 0.7 0.9
1.5±1.1 9.7±15.8 85.0±123.1 173.2±222.8 337.3±363.7 478.5±450.6 612.7±516.2
Fraction of Fraction of highcost edges Steiner proteins 0.034 0.058 0.273 0.345 0.389 0.404 0.407
0.471 0.295 0.248 0.233 0.213 0.198 0.188
2500
Nb. proteins
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1500
1000
500
0 0.5
1
1.5
2
2.5
3
Average of prizes
Fig. 1. This ﬁgure shows the strong correlation between the average node prize in each dataset (xaxis) and the number of proteins found in the inferred tree (yaxis). Diﬀerent types of points correspond to diﬀerent values of λ: vertical crosses λ = 0.05, diagonal crosses λ = 0.1, stars λ = 0.2, empty boxes λ = 0.3, ﬁlled boxes λ = 0.5, empty circles λ = 0.7, ﬁlled circles λ = 0.9. Note the linearity of the relation for each given value of λ.
values of λ. To assess the quality of the trees found, we computed the average number of shared GO Slim annotations between neighbors, and compared it to random trees, either weighted or not (Fig 3). the average number of common annotations is higher for low values of λ (Fig 3), showing a clear functional enrichment of the Steiner trees. Topology and PIN weights only account for a part
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Fig. 2. This tree is formed by the superposition of all 56 Steiner trees found for λ = 0.2. Link intensity is proportional to the number of times the link was found, either in one sense or another; in case of links inferred in diﬀerent directions, the orientation represented is the one mostly found. Links found in less than 30% of the trees are not shown for clarity. Grey nodes represent the proteins involved in the known pheromone pathway.
of this enrichment, shown by the simulations with random weighted trees, the rest being a combined consequence of both the proteins and the paths selected in the tree, which can thus be considered to represent biologically meaningful transduction networks. For high values of λ, this enrichment is not visible, and we will therefore focus on results at low λ. Previous to analyses, a technical bias has to be accounted for. Due to diﬀerences in inbetweenness – or connectivity, see [11] –, certain proteins occur more or less often in the Steiner trees. Indeed, proteins with a high inbetweenness in the PIN tend to be frequently present in the Steiner trees, even if they are attributed a low prize. From a probabilistic point of view, including these proteins in the Steiner tree allows to gain access to proteins with a positive contribution to the global tree cost, enough to compensate for their own relative costs. One can see this trend in Fig 4: proteins selected more often have a high inbetweenness. Still, this correlation is only partial (R2 = 0.37), and let ample space for other factors to explain presence of certain proteins in the ﬁnal trees.
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Fig. 3. Histogram of the average number of shared GO Slim annotations per link, on average on all 56 inferred trees at a given value of λ. First histogram represents the real values, second random weighted trees, and third random unweighted trees (See Mat. Meth.). Note the high diﬀerences for low values of λ. Table 2. Properties of the 11 putative Steiner proteins found the most frequently in λ = 0.2 datasets. k stands for connectivity and ”Inbet.” for inbetweenness. Gene name Protein name Frac. found Frac. found Ratio k Inbet. (x105 ) (real data) (random data) YBR160W YDR388W YHL048W YFL039C YER118C YJR091C YCL040W YBR159W YGL181W YPL181W YMR059W
CDC28 RVS167 COS8 ACT1 SHO1 JSN1 GLK1 IFA38 GTS1 CTI6 SEN15
0.66 0.52 0.45 0.45 0.43 0.43 0.43 0.41 0.41 0.41 0.34
0.57 0.31 0.09 0.17 0.06 0.44 0.003 0.09 0.09 0.02 0.007
1.2 1.7 5.0 2.7 7.0 1.0 144 4.8 4.5 20.3 47.5
227 121 46 47 42 293 6 101 43 26 57
15 4.9 0.63 1.1 1.5 25 0.25 1.9 1.5 0.26 1.4
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1e+07
Inbetweeness
1e+06
100000
10000 1
10
100
1000
Nb. Steiner
Fig. 4. Number of times a protein appears as Steiner (in all trees inferred) vs inbetweenness of the protein in the PIN. Note the correlation between them.
An interesting feature of our formalism is the deﬁnition of Steiner proteins, i.e proteins present in the Steiner trees without being highly diﬀerentially expressed. These proteins form bridges between groups of proteins with a positive contribution to the optimization criterion, and they could not be discovered by analyzing only the expression levels in the microarray, as they do not diﬀer signiﬁcantly from the background. Is is the combination of information from the PIN structure and expression data that unveil them. In the following analyses we focus on the Steiner proteins that appear at low values of λ, i.e those less distinguishable from the background expression. In order to quantitatively measure the signiﬁcance of Steiner proteins, we did a bootstrap experiment by generating Steiner trees for random expression data, obtained by permutations of the real datasets. Then, we compared the frequency of occurrence of proteins found very often as Steiner proteins in the real data to their frequency of occurrence in this randomized data; the ratio of these quantities was then used to assess the biological signiﬁcance of the putative Steiner proteins (see Table 2), a high ratio meaning biologically meaningful inference and a low one typical of an artifact due to PIN topology and high inbetweenness bias. Proteins with such a high ratio have an average inbetweenness (see Table 2). Using this table, one can easily see that the proteins CDC28, JSN1 and RVS167 should not be accounted as Steiner proteins, based on the ratio value and their very high inbetweenness. To get better insights about these proteins and their
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Fig. 5. Main ﬁrstorder interactions of the proteins identiﬁed as Steiner proteins, λ = 0.2. Steiner proteins are shown in grey. Link intensity is proportional to the number of times the link is found when protein is considered as Steiner. Only links found in more than 50% runs are shown.
implication in the pheromone response as Steiner proteins, we looked which partners they interact with. The partners found in more than half of the trees are represented in Fig 5. Many interactants are membrane proteins, in particular PRM proteins [28]. One interesting feature is that the Steiner proteins COS8 and SHO1 seem to be strongly interacting, as ACT1 and GLK1 do either. We detail these two cases in the following paragraphs. GLK1 give access to the FIG1 protein, a membrane protein which has already been implicated in the pheromone response and in particular in cell fusion [29]. Another protein implicated in the glucose metabolism, GTS1, is inferred as a Steiner protein. As both these proteins are interacting with the actin protein ACT1, one could hypothesize a crosstalk between pheromone response, glucose metabolism and cytoskeleton structure. If role of the cytoskeleton in mating is quite wellknown, implication of the glucose metabolism is not, but could be a sign of a global regulation of the cellular state previous to mating, as GTS1 is also a known regulator of transcription. COS8 is found at the end of the SST2SHO1 cascade. SST2 is the regulator of desensitization of the pheromone pathway, while SHO1 is the main cell osmosensor and initiates various signaling pathways. The subtree found behind COS8 is composed of membrane proteins and the fattyacid elongase ELO1. Moreover, COS8 interacts often with proteins involved in sphingolipid synthesis, such as LAC1 and AUR1 (not shown in Fig 5 because they occur less than 50% of the time). Finally, the main cascade leading to IFA38, a betaketo reductase implicated in fatty acid metabolism and also found as Steiner protein, is indeed passing by COS8. The multiple interactions of COS8 with these proteins, either membranespanning or located in the ER, allows to hypothesize that COS8
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plays a role in the secretory pathway – probably in relation with sphingolipid synthesis– during pheromone response. Interestingly, COS8 is one but a member of a very conserved gene family [30], and ﬁnding the function of COS8 could help to understand the role of the entire family.
4
Discussion
In this work we presented an eﬃcient strategy to infer structure relations from sparse gene expression information and proteinprotein interaction probabilities. This approach is based on statistical physics principles, is scalable (completely parallelizable) and is expected to be wellsuited for large networks. The scheme is highly eﬃcient (the computation time scales as DE where E is the number of edges of the protein network, and it normally suﬃces to take D = O(logN ) to achieve optimal values). This property allowed us to explore values of the parameter λ and a large number of pathways very quickly, much faster than it would have taken with complete algorithms and other available heuristics. The main drawback of the approach resides in its input limitations, that is, it cannot infer new interactions between proteins and must follow the structure of the PIN given as input. This makes it diﬃcult to apply our method to organisms where the PIN is unknown or poorly described, which is the case for many organisms, such as human. However, this issue is becoming obsolete with the rise of new experimental techniques in the proteomic ﬁelds. Moreover, there are bioinformatic solutions that could be used in order not to be limited by this problem. First, one could add in the PIN very high cost edges on a set of putative interactions. Analysis of the Steiner trees with increasing values of λ, may allow to see, among the added edges, which are selected more frequently by the algorithm, and thereby discriminate between them. Second, as the methods to infer proteinprotein interactions based solely on sequence become more eﬃcient (see e.g. [31]), it should be possible to develop an integrated framework where protein interactions are inferred numerically before applying our methodology. Our methodology, while using stateoftheart computational techniques, is able to infer quantitatively which Steiner proteins could play a role in a given context, as represented by expression data. The network representation allows a clear interpretation: speciﬁc interactions are predicted in deﬁned conditions. This type of predictions is easy to conﬁrm experimentally by doublehybrid and genetic experiments, making our methodology an invaluable input for wet labs. Collaborations have indeed been started to experimentally test our predictions. Moreover, our algorithm could be made still more eﬃcient, by including genetic or regulatory interactions in the base network or searching for protein complexes instead of protein interactions. Developments in this sense, coupled with experimental validations of our predictions, will ﬁnally allow the development an integrated messagepassing framework for systems biology, in direct contact with experimental data and labs.
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Acknowledgements We thank S. Fortunato for the computation of the inbetweenness, and J.M. Fran¸cois for help in the biological analyses. M.B.B and A.B acknowledge a fellowship from Politecnico di Torino and a combined Microsoft research/Politecnico di Torino funding.
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Formal Analysis of the Genetic Toggle Giampaolo Bella1 and Pietro Li` o2 1 Dipartimento di Matematica e Informatica, Universit` a di Catania, Viale A. Doria 6, 95125 Catania, Italy
[email protected] 2 Computer Laboratory University of Cambridge 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK
[email protected] Abstract. The formal analysis of the toggle switch, which is among the most common motifs of genetic networks, shows that along with the powerful development of mathematical modelling, formal methods can be of great help in investigating the properties of genetic networks. In particular, a general approach to modelling genetic networks through the language of higherorder logic is advanced and mechanised in the theorem prover Isabelle. An inductive deﬁnition provides a formal model for the genetic toggle as the set of all possible evolutions of such network. Gene polymerase and protein concentration are formalised as primitive recursive functions. The main properties of the genetic toggle are conﬁrmed upon the model: it is possible that one protein exceeds a stated concentration threshold and the other protein does not; it is impossible that both proteins exceed their respective concentration thresholds at the same time. To the best of the authors’ knowledge, this is the ﬁrst contribution of theorem proving in the area of genetic network analysis, and as such may set the foundations for a new niche of research.
1
Introduction
Genes function in highly interconnected, hierarchical, and nonlinear chemical networks. Representing interactions between biological molecules as a network provides us with a conceptual framework that allows us to identify general principles that govern these complex systems. A network is best represented as a graph that is made up of nodes, which denote the components, and links, which denote the interaction between the components. In a genetic or transcriptional regulatory network, nodes represent DNA binding proteins or target genes and directed edges represent regulatory interaction where the protein regulates the expression of the target gene. A meaningful approach to investigate biological networks is the identiﬁcation of network motifs. Motifs can be deﬁned as a set of network nodes with speciﬁc molecular functions which are arranged together and perform some ’useful’ processes. The behaviours of motifs are generally not separable from the rest of the system and they constitute only part of a recognizable systems level function. There are several known motifs in genetic networks for example toggle switches, amplitude ﬁlters, oscillators, frequency ﬁlters, noise P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 96–110, 2009. c SpringerVerlag Berlin Heidelberg 2009
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ﬁlters and ampliﬁers, combinatorial logic, homeostats, rheostats, logic (e.g. see [1,2,3,4]). The ﬁrst step of the analysis of a network motif is to model the process of gene expression, i.e. the process through which the RNA polymerase produces mRNA molecules from a DNA template, and the process of protein synthesis, which is the formation of proteins from an mRNA template. The mathematical description of the variation of biomolecular concentrations may be based on a set of diﬀerential equations (ODE) or on stochastic simulations by means of the Gillespie algorithm [5]. The diﬀerential equations approach is eﬀective in describing the average behaviour of the reaction set and allows to incorporate non linearities; instead, the stochastic approach provides a more realistic approach because, in many cases, in the cellular environment, the number of molecules tend to be small and concentrations may ﬂuctuate. For example, using the ODE approach, the rate of change of mRNA (R) and of protein (P) concentrations depend on the formation λ and the degradation δ rates and can be written as: dR = λR − δR R ; dt
dP = λP R − δP P dt
The steadystate solutions of the equations are: ˜ = λR ; R δR
λP λR λP P˜ = = δP R δR δP
The most important and ubiquitous motif network is the toggle switch, which consists of two genes, each of which produces a protein that represses the transcription of the other gene. The toggle can be described by a system of coupled nonlinear diﬀerential equations, which are usually very diﬃcult to solve exactly, leaving us to two choices: either solve the system numerically or ﬁnd some kind of approximate solution. A stochastic implementation always results in intensive simulations. If we consider a network with a large number of genes, the mathematical analytical treatment becomes impossible and the stochastic simulation computationally very intensive. Therefore, methods that provide a diﬀerent, complementary understanding of the behaviour of biological networks are greatly needed, and formal methods are good candidates. Some of them come with compelling computer support, which turns out particularly useful also when reasoning about biological problems. For example, this is the case with the use of model checking techniques in this area, a synergy that has produced signiﬁcant ﬁndings in the last years. A nonexhaustive account on such ﬁndings would have to mention the seminal paper where a genetic network is simpliﬁed into a ﬁnitestate machine amenable to model checking [6]. Then, the account should cover the ﬁrst complete application to a paradigmatic case study, the genetic toggle, where the language of lineartemporal logic is used to express the main properties of the network [7]. There are also more recent developments (omitted here due to space limitations). Theorem proving is another popular computerassisted technology to exceed the limitations of penandpaper reasoning. Its comparison to model checking, which lies beyond our focus, has notoriously been the subject of vast research.
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In brief, model checking excels for the limited amount of human intervention, which makes it particularly appreciated in the industrial world, but can only handle systems of limited size [8]. By contrast, theorem proving typically reaches outstanding levels of detail in the expression of a system of unbounded size and of its properties, but requires more human intervention [9]. The main contribution of this paper is to lay the ground for the use of theorem proving in the ﬁeld of biological problems such as genetic network analysis. To our knowledge, the potentialities of theorem proving in improving our understanding of genetic networks have never been explored before. Our ﬁndings are promising. A genetic network is formalised by mathematical induction as the set of all its possible evolutions, which in the real world are determined by stochastic factors. Such factors may safely be abstracted away because, whatever network evolution they may cause, that evolution is certain to belong to the inductive model of the genetic network. Invariant properties of the model therefore apply to the real network. Our inductive model of a genetic network is qualitative as it provides an operational semantics of the network. In particular, gene polymerase and protein concentration are deﬁned by primitive recursion. All deﬁnitions are fed to the interactive theorem prover Isabelle [9], which supports the veriﬁcation of properties of the model. This paper studies the genetic toggle to demonstrate the accuracy and expressiveness of the approach we propose. The organisation of this manuscript is simple. A very brief outline of Isabelle (§2) precedes the description of our approach to formally analysing genetic networks (§3). Then, the genetic toggle is modelled and veriﬁed as a case study (§4). Some concluding remarks terminate the presentation (§5).
2
Isabelle
Isabelle is a generic, interactive theorem prover. Generic means that it can reason in a variety of formal systems. This paper refers to Isabelle/HOL [9], which supports the formal language higherorder logic, a typed formalism that allows quantiﬁcation over functions, predicates and sets, but has no temporal operators. Interactive means that it is not entirely automatic and, rather, requires a good amount of human intervention. But Isabelle also provides much automation. Its simplifier, which can be invoked by the proof method simp, combines rewriting with arithmetic decision procedures. Its automatic provers can solve most simple proof scenarios. For example, the proof method blast implements a fast classical reasoner, and auto combines that with the simpliﬁer. Proofs are conducted interactively. In a typical proof, the user directs Isabelle to perform a certain induction and then to simplify the resulting subgoals. Any surviving subgoals may be given to an automatic prover or be reduced to other subgoals by means of some lemma. Failure to ﬁnd a proof for a conjecture may simply mean that the user is not skilled enough; otherwise, it may exhibit what in the modelled system contradicts the conjecture and hence help in locating a system bug or an erroneous human intuition. The command list used to prove a theorem can be seen as a proof sketch. Conﬁdence that the proof is sound comes from inspecting the line of reasoning adopted and the lemmas it requires.
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Formal Analysis of Genetic Networks
Our approach to modelling genetic networks is inspired to an existing model of security protocols [10], and is mechanised in the theorem prover Isabelle. An inductive deﬁnition provides an operational semantics for the computer network underlying the distributed execution of a security protocol. The computer network is qualitatively seen as the interaction of agents who exchange the protocol messages, and hence our idea to view a genetic network as the interaction of genes and proteins who exchange some signals. Our presentation proceeds by deﬁning the basics of a general model of genetic networks, which can then be used to tackle any case study. Due to space limitations, the subsidiary lemmas about the general model, which, as is typical with theorem proving, assess its correctness, must be skipped. Likewise, a didactic example of how to model a simple network cannot be presented, but an example of realistic size is given in the next Section. Following the observation that the main agents of a genetic network are genes and proteins, they are introduced as free types, that is as type declarations: typedecl gene typedecl protein
We model four main events that may take place in a genetic network. As a start, what a gene may do is to code a protein. A protein may in turn either trigger or inhibit a gene expression. Proteins may also diﬀuse in the cell volume and be degradated (misfolding and ubiquination) before they succeed in regulating a gene. These four events are respectively formalised by the four keywords Produces, Triggers, Inhibits and Degrades, which are introduced as type constructors of the datatype event. It can be seen how each type constructor takes two freetypes as parameters, except for the protein degradation event, which, as expected, only takes one: datatype event = Produces gene protein  Triggers protein gene  Inhibits protein gene  Degrades protein
For example, the event Triggers A b is deﬁned in our language, indicating the act of a protein A that induces a gene b to express its protein. The details or intermediate steps underlying each event, such as transcription to RNA, are hidden behind each event. An operational semantics only expresses what event takes place rather than specifying how each event is carried out. The datatype may of course be extended in the future to capture additional network events. We continue by modelling the main condition for a gene to express a protein, that is the gene level of polymerase. It is assumed that all genes have some positive polymerase level since the very beginning of the observation, so that they can begin expressing their proteins. Such initial polymerase is introduced as constant for all genes for simplicity: consts initialpolymerase :: int
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Specifying diﬀerent initial polymerases for the genes would be trivial as the constant just introduced would gain an extra parameter of type gene. The requirement that the initial polymerase is strictly positive can be easily introduced as an axiom of our model: axioms initialpolymerase_value [iff]: initialpolymerase > 0
The [iff] declaration tells both the simpliﬁer and the classical reasoner of Isabelle that axiom named initialpolymerase value can automatically be appealed to in solving the subgoals that may arise from subsequent proof attempts. For example, it may lead a set of facts to a desired absurd when the proof strategy proceeds from the falsiﬁcation of the thesis. The polymerase clearly is a dynamic value. It is inﬂuenced by the events that proteins carry out, which are triggers or inhibitions, as deﬁned above. Therefore, such induced polymerase can be easily declared as: consts inducedpolymerase :: gene ⇒ event list ⇒ int
The second parameter introduces the list of events that have taken place, which modify the induced polymerase. It may be currently obscure where such a list comes from, but the rest of the treatment will show how it can be eﬀectively deﬁned by mathematical induction. The function modelling the induced polymerase can be deﬁned by primitive recursion on the structure of the list of events that is considered. It is useful to remind that Isabelle’s lists are built from right to left, that [] denotes the empty list, and that # is the list cons operator: primrec inducedpolymerase x inducedpolymerase x Produces y Y ⇒  Triggers Y y ⇒
[] = 0 (head#rest) = (case head of inducedpolymerase x rest if x=y then inducedpolymerase else inducedpolymerase  Inhibits Y y ⇒ if x=y then inducedpolymerase else inducedpolymerase  Degrades Y ⇒ inducedpolymerase x rest )
x x x x
rest + 1 rest rest  1 rest
The primitive recursive deﬁnition consists of two equations. The ﬁrst, which is for the base case, is very simple as it states that the polymerase that an empty list of events induces upon a gene x is zero. Intuitively, because no events are available in this case, they induce no variation upon the polymerase. The second equation is the inductive step, as it considers a list of events of which the head event head is emphasized. The equation contains a case analysis deriving from each of the four possible events that may get instantiated as head. The ﬁrst and last line of the case analysis respectively make sure that a gene expression of a protein and a protein degradation do not change the polymerase that the rest of the list rest induces upon the generic gene x. To describe it with the terminology of term rewriting, the expression inducedpolymerase x (head#rest) is evaluated as inducedpolymerase x rest, conﬁrming that the head event just cancels itself out during the symbolic evaluation.
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The two internal lines, respectively Triggers Y y and Inhibits Y y, are perhaps more interesting. Let us begin by studying the former. It resolves into an ifthenelse expression whose condition is that the generic gene y that the event considers be exactly the gene x whose induced polymerase is being evaluated. Only if this check is true does the evaluation resolve into inducedpolymerase x rest + 1, which in our qualitative approach is to be interpreted as a generic increase of the polymerase that the rest of the list rest induces upon gene x (of course, a symbolic parameter might be used instead of the unit). Otherwise, the symbolic evaluation just skips head, returning inducedpolymerase x rest. The other line can be interpreted similarly, but an Inhibits Y y event should induce a decrease over the polymerase of the gene, and this is expressed by the  1 (or minus some parameter instead). Having deﬁned the initial and the induced polymerase, the current polymerase can be introduced as a sum of the two. The declaration must account for two parameters due to the induced polymerase. The deﬁnition is simple, and can be introduced compactly by Isabelle’s constdefs command as follows: constdefs currentpolymerase :: gene ⇒ event list ⇒ int currentpolymerase x nt == initialpolymerase + inducedpolymerase x nt
It can be seen that the current polymerase has a static part, which was available initially, and a dynamic part induced by the events in the considered list. The codomain of our functions for the polymerase is int rather than nat, which might have been more intuitive. Our choice is determined by the primitive recursive deﬁnition of the induced polymerase seen above, which decreases the current value when an appropriate inhibition event is evaluated. For example, the polymerase induced over a gene by a list of inhibition events on that gene is a negative value. It may be interpreted in the model as extra incapacity to express the protein. In consequence, also the other two polymerase functions must return an integer to avoid type clashes. Alternatively, the deﬁnition of the induced polymerase should have accommodated the extra case analysis that protein inhibition decreases the induced polymerase if and only if it is not already zero. We made the simpler modelling choice. Turning our focus to proteins, the concentration appears to be the main prerequisite to a protein activity. A protein typically begins to inhibit or trigger genes depending on whether its concentration exceeds some threshold. The same mechanisms used above can be adopted here to deﬁne the concentration: consts concentration :: primrec concentration X concentration X Produces y
protein ⇒ event list ⇒ int
[] = 0 (head#rest) = (case head of Y ⇒ if X=Y then concentration X rest + 1 else concentration X rest  Triggers Y y ⇒ concentration X rest  Inhibits Y y ⇒ concentration X rest
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⇒ if X=Y then concentration X rest  1 else concentration X rest
)
The deﬁnition insists that the concentration be initially zero. The inductive step provides the symbolic evaluation of the concentration of a generic protein X over a list of events whose head event is head. The ﬁrst line of the case analysis states that when head exactly is a production event of protein X, then the concentration increases by being evaluated as concentration X rest + 1. The last line follows the same pattern to make sure that when the protein degrades its concentration decreases. The two central lines leave the symbolic evaluation unaltered by skipping the head event. A ﬁrst version of this deﬁnition decreased the concentration also in the cases of trigger or inhibition events to signify that a protein that binds to a promoter of a gene is no longer available. However, it was later preferred to simplify it as it now stands with a negligible overhead: making sure that a trigger or inhibition event is always followed by a degradation event. This proviso must be accounted for when deﬁning the network, as we shall see below. The remark made about the int codomain also applies here: it is a technical requirement of our formalisation. Although no negative concentration exists in the real world, it may be interpreted in the model as extra distance from a required threshold. Alternatively, it may be removed by adding a case study to the symbolic evaluation of the concentration in case of a degradation event. Note that only one function suﬃces to deﬁne the concentration because we are assuming that the concentrations of all proteins be initially zero. Removing this assumption would be costless, resembling the pattern of the deﬁnitions introduced for the polymerase, that is to introduce an initial value, then an induced value and ﬁnally the current value.
4
A Case Study: The Genetic Toggle
The toggle switch is a cyclic digenic system (say genes g1 and g2 ) with negative feedbacks between the genes through the negative regulation of their proteins. A simple way to model the regulatory control of a protein on a gene is to modify R by considering the binding properties of the protein on a gene, σ1 or σ2 . In a toggle switch network we would expect the system to exhibit two mutually exclusive behaviours  either g1 is high, keeping expression of g2 low, or conversely, g2 is high, keeping expression of g1 low. The toggle can switch from one stable state to the other by means of an external signal. Noteworthy, the toggle is at the core of the synthetic biology [11]. For simplicity, we assume that the switch is composed of symmetric elements so that the rate constants are identical for each half of the network and the mathematical model takes the form: dR1 = λR σ1 − δR R ; dt
dR2 = λR σ2 − δR R dt
dP1 = λP R1 − δP P1 ; dt
dP2 = λP R2 − δP P2 dt
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Here, the λs and δs respectively are formation and degradation parameters. The numerical solution of this set of equations provides the determination of the two steadystate thresholds of proteins P1 and P2, which depend upon one another (see also [1]). The set of equations identiﬁes curves that are called nullclines and their intersection is called equilibrium point. Although recent methodological developments have made it possible to estimate rate constants from experimental observations such as microarray data [12], from a qualitative perspective the identiﬁcation and characterization of the presence of a toggle switch may come from the formal analysis of the behaviour of the system, as we shall see below. 4.1
Modelling the Genetic Toggle
The general model of genetic networks introduced above can be demonstrated upon the genetic toggle. Few additional deﬁnitions are necessary for this speciﬁc example. As our approach is qualitative, we abstract away the formation and degradation parameters seen above. The pair of threshold concentrations of a protein can be introduced as a function parameterised over the integers and the proteins: consts threshold :: nat ⇒ protein ⇒ int axioms threshold_1A_less_2A [iff]: threshold 1 A < threshold 2 A threshold_1B_less_2B [iff]: threshold 1 B < threshold 2 B threshold_1A_pos [iff]: threshold 1 A > 0 threshold_1B_pos [iff]: threshold 1 B > 0
By having the integers (rather than the booleans) as a parameter, our declaration also supports more than two thresholds. The axioms formalise the typical requirements for this case study. We deﬁne the total concentration of proteins that can be produced as the sum of the two higher thresholds. Because we omit naming the function deﬁnition, Isabelle assigns the default name totalproteins_def. constdefs totalproteins :: int [iff]: totalproteins == threshold 2 A + threshold 2 B
The model for the genetic toggle can be now deﬁned inductively as follows: inductive set network :: event list set where base:
[] ∈ network
 aprA:
[[ nt1 ∈ network; currentpolymerase a nt1 > 0; concentration A nt1 + concentration B nt1 < totalproteins ]] =⇒ Produces a A # nt1 ∈ network
 bprB:
[[ nt2 ∈ network; currentpolymerase b nt2 > 0; concentration A nt2 + concentration B nt2 < totalproteins ]] =⇒ Produces b B # nt2 ∈ network
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 Ainb:
[[ nt3 ∈ network; concentration A nt3 ≥ threshold 1 A ]] =⇒ Degrades A # Inhibits A b # nt3 ∈ network
 Aina:
[[ nt4 ∈ network; concentration A nt4 ≥ threshold 2 A ]] =⇒ Degrades A # Inhibits A a # nt4 ∈ network
 Bina:
[[ nt5 ∈ network; concentration B nt5 ≥ threshold 1 B ]] =⇒ Degrades B # Inhibits B a # nt5 ∈ network
 Binb:
[[ nt6 ∈ network; concentration B nt6 ≥ threshold 2 B ]] =⇒ Degrades B # Inhibits B b # nt6 ∈ network
 dgrA:
[[ nt7 ∈ network; Produces a A ∈ set nt7 ]] =⇒ Degrades A # nt7 ∈ network
 dgrB:
[[ nt8 ∈ network; Produces b B ∈ set nt8 ]] =⇒ Degrades B # nt8 ∈ network
The base case sets the base of the induction. The last two inductive rules model a stochastic degradation of a protein, as they may ﬁre any time provided that the protein was produced. The remaining rules model the distinctive features of the genetic toggle. Rules aprA and bprB produce proteins A and B respectively upon condition that the polymerase of the involved gene is positive and that the total concentration of proteins has not been reached. Then, four inhibition rules follow, all with the same structure enforcing the proviso that a protein degrades after it links to a gene. Precisely, rules Ainb and Bina state that a protein inhibits the other gene when the protein concentration exceeds its ﬁrst threshold. Rules Aina and Binb state that a protein inhibits its own gene when the protein concentration exceeds its second threshold. 4.2
Verifying the Genetic Toggle
As a start, following the axioms deﬁning the thresholds, we can prove that also the second thresholds are positive: lemma threshold_2A_pos: threshold 2 A > 0 apply (blast intro: less_trans) done lemma threshold_2B_pos: threshold 2 B > 0 apply (blast intro: less_trans) done
These lemmas are sometimes useful below. Each can be proved by a single invocation to the classical reasoner, the blast method, with an appeal to lemma less_trans, which conﬁrms the transitivity of the less relation. Having observed the model of the genetic toggle, our ﬁrst conjecture was that whenever a gene produces a protein, the polymerase of the gene must be positive. This can be formalised as follows: lemma aprA_cp: [[ Produces a A ∈ set nt; nt ∈ network ]] =⇒ currentpolymerase a nt > 0
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Precisely, the statement insists upon two preconditions: one is that nt is a network list of the model toggle network ; the other one is that the event Produces a A appears in nt. The conclusion is that the current polymerase of gene a in the network list nt is positive. Note Isabelle’s meta implication “=⇒”, not to be confused with objectlevel implication “−→”. We begin proving this conjecture, which is the goal of our proof attempt, by bringing the main premise into the inductive formula. This can be done by Isabelle’s method erule, which applies resolution of the goal with the stated theorem, reverse modusponens rev_mp in this case, and then eliminates the preconditions that were necessary to the application: apply (erule rev_mp)
As a result, Produces a A −→ currentpolymerase a nt > 0 becomes the conclusion of the proof goal. Structural induction can be applied now, which precisely is the resolution of the current proof goal with the network.induct theorem that Isabelle builds automatically to reﬂect the inductive structure of constant network (it is very long but can be viewed by interacting with Isabelle): apply (erule network.induct)
Our goal is now split up into 9 subgoals, one per each rule deﬁning network. We can apply the simpliﬁer to all goals by the method simp_all but three subsidiary lemmas proved earlier (omitted here) must be invoked as rewrite rules for symbolic evaluation of the current polymerase function over network events. Note that currentpolymerase_Triggers is unnecessary because the genetic toggle features no trigger events. apply (simp_all add: currentpolymerase_Produces currentpolymerase_Inhibits currentpolymerase_Degrades) oops
By inspecting the resulting proof state, we realise that the proof cannot be terminated and hence the oops command tells the prover to leave it and proceed. The proof state, consisting of two subgoals, can be analysed from Figure 1. In general, the inability to prove a conjecture may be due to insuﬃcient skill of the human analyser, or to falseness of the conjecture. The proof state conﬁrms that the latter applies here. It can be seen that the conclusion of each subgoal cannot be proved upon the available preconditions — the second one in each subgoal being the inductive hypothesis. The subgoals take our attention upon the rules that cause them, Aina and Bina, which are the only rules that inhibit the very gene a mentioned in the conjecture and therefore decrease its polymerase. Hence, the subgoals highlight two counterexamples to the conjecture: one is a network list where the polymerase is positive and then a is inhibited by ﬁring of the rule Aina ; the other one is analogous but involves ﬁring of the rule Bina. So, we decided to weaken the conjecture by involving only network lists whose last event (i.e. the head event) is a gene production. Using the predeﬁned function hd to extract the head of a list, this can be formalised and proved as follows:
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Fig. 1. The ﬁnal proof state of a conjecture lemma aprA_cp: [[ Produces a A = hd(nt); nt ∈ network ]] =⇒ currentpolymerase a nt > 0 apply (erule rev_mp) apply (erule network.induct) apply (simp_all add: currentpolymerase_Produces) apply (simp add: currentpolymerase_def) done
The proof terminates, so that the conjecture holds. The script needs an extra application of the simpliﬁer to solve the subgoal deriving from rule base, which is Produces a A = hd [] −→ 0 < currentpolymerase a [], because the prover needs to evaluate the head of an empty list, unlike in the previous proof attempt. However, this subgoal can be solved straightforwardly by unfolding the deﬁnition of the current polymerase, which obtains the required postcondition by positiveness of the initial polymerase (axiom initialpolymerase_value, §3). Moreover, only the rewrite rule currentpolymerase_Produces is necessary in the main simpliﬁcation step because the theorem precondition Produces a A = hd(nt) only requires symbolic evaluation of the current polymerase over production events, as it rules out all other events. Our primary aim is to prove the main property of the genetic toggle, which is bistability. We address as bistabilityA the situation in which protein A exceeds its second threshold but protein B does not; as bistabilityB the opposite situation where protein B exceeds its second threshold but protein A does not; and as instability the situation in which both proteins exceed their respective second thresholds. Therefore, proving bistability evaluates to proving that:
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the network may evolve to bistabilityA; the network may evolve to bistabilityB; the network may not evolve to instability. Isabelle is best tailored to proving safety properties, that is invariants of all network lists. Hence, we set about the ﬁrst two properties by attempting to prove that they fail to hold, that is that the network may not evolve to either bistabilityA or bistabilityB. The prover will exhibit the counterexamples we are looking for, if these exist. Here is the conjecture that bistabilityA may not be reached and its (failed) proof attempt: theorem no_bistabilityA: nt ∈ network =⇒ ¬ (concentration A nt ≥ threshold 2 A ∧ concentration B nt < threshold 2 B) apply (erule network.induct) apply (insert threshold_2A_pos) apply auto oops
After applying induction, lemma threshold_2A_pos is inserted in the ﬁrst subgoal, which arises from rule base — simplifying that subgoal without inserting that lemma would leave threshold 2 A ≤ 0 −→ ¬ 0 < threshold 2 B. Then,
Fig. 2. Proving that bistabilityA may be reached
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Fig. 3. Proving that bistabilityB may be reached
the auto method leads to a proof state, shown in Figure 2, that cannot be terminated. It features subgoals arising respectively from rules where protein A is produced (aprA ) or B degraded (Bina, Binb and dgrB ). Each subgoal describes one of the counterexamples we were seeking to establish that bistabilityA may be reached. For example, the second subgoal describes a network list where A ’s concentration exceeds its second threshold, while B ’s concentration equals its second threshold, and hence the ﬁring of rule Bina, where B degrades, clearly falsiﬁes the main conjecture. The reasoning about bistabilityB is analogous. Here is the conjecture that bistabilityB may not be reached and its (failed) proof attempt: theorem no_bistabilityB: nt ∈ network =⇒ ¬ (concentration A nt < threshold 2 A ∧ concentration B nt ≥ threshold 2 B) apply (erule network.induct) apply (insert threshold_2B_pos) apply auto oops
The ﬁnal proof state, shown in Figure 3, cannot be terminated. It has subgoals arising respectively from rules where protein B is produced (bprB ) or A degraded
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(Ainb, Aina and dgrA ). Each subgoal describes one of the counterexamples we were seeking to establish that bistabilityB may be reached. For example, the ﬁrst subgoal describes a network list where A ’s concentration is less than its second threshold, while B ’s concentration equals its second threshold minus 1, and hence the ﬁring of rule bprB, where B is produced, clearly falsiﬁes the main conjecture. The ﬁnal step is to prove that instability is impossible as: theorem no_instability: nt ∈ network =⇒ ¬ (concentration A nt > threshold 2 A ∧ concentration B nt > threshold 2 B) apply (erule network.induct) apply (insert threshold_2A_pos) apply auto done
5
Conclusions
We have formally analysed the genetic toogle by theorem proving and established its main correctness property, bistability: proteins may prevail against each other, but never at the same time. The toggle had already been studied by model checking [7]. This method simpliﬁes the toggle as a ﬁnite state machine so that the model checker can exhaustively search the state space, whose size is inversely proportional to the speed of the search. A temporal logic is used to express the property that a protein may prevail on some path in the state space. Our approach requires more intervention to the human analyser but scales up to systems of unbounded size. Not only is this claim supported by experience from other application areas [10], but also by the very nature of an inductive proof. The theorem prover only helps the human check a conjecture against each inductive rule deﬁning the model. Also, the language of higherorder logic implemented in Isabelle/HOL is most expressive for the properties of the toggle. Our contribution therefore is the ﬁrst investigation about the use of theorem proving to formally analysing genetic networks. The motivation for taking this approach lies in the mathematical complexity of even small genetic networks. Noteworthy, besides genetic (i.e. transcriptional regulatory) networks, many different types of biological networks exist, such as the protein interaction networks where nodes represent proteins and links represent physical interaction between the proteins, and metabolic networks where nodes represent small molecules and links represent direct enzymatic conversion between the small molecules. The mathematical modeling of such interaction heterogeneity may scale up quickly with the number of components and of interaction types, but formal methods oﬀer a complementary, complexityreducing approach. Moreover, formal methods have a dialectic relationship with mathematical methods. Highly automatic methods based on model checking may take the analysis of genetic networks well beyond the world of mathematicians, for example by reaching industrial worlds where such methods are widely adopted. Typically, the human launches the checker on a property and waits for its output, which is only either positive or negative. This paper makes the case for
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highly expressive theorem proving, which is expected to open the ﬁeld of genetic network analysis to the world of pure logicians. Typically, the human develops the proof of a property with the help of the prover, and then anyone can inspect and selfevaluate that proof. The formal analysis of biological problems has only just begun. Acknowledgments. The authors gratefully acknowledge funding for reciprocal visits at their respective Institutions from the British Council’s BritishItalian partnership programme for year 2008, grant entitled “Computerassisted veriﬁcation for safety properties of genetic networks underlying liver regeneration”.
References 1. YegerLotem, E., et al.: Network motifs in integrated cellular networks of transcriptionregulation and proteinprotein interaction. Proc. Natl. Acad. Sci. U.S.A. 101, 5934–5939 (2004) 2. Tyson, J.J., Chen, K.C., Novak, B.: Sniﬀers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr. Opin. Cell Biol. 15, 221–231 (2003) 3. Ninfa, A.J., Selinsky, S., Perry, N., Atkins, S., Xiu Song, Q., Mayo, A., Arps, D., Woolf, P., Atkinson, M.R.: Using twocomponent systems and other bacterial regulatory factors for the fabrication of synthetic genetic devices. Meth. Enzymol. 422, 488–512 (2007) 4. Atkinson, M.R., Savageau, M.A., Myers, J.T., Ninfa, A.J.: Development of genetic circuitry exhibiting toggle switch or oscillatory behavior in Escherichia coli. Cell 113, 597–607 (2003) 5. Gillespie, D.: Exact simulation of coupled chemical reactions. Journal of Chemical Physics 81, 2340–2345 (1997) 6. Batt, G., de Jong, H., Geiselmann, J., Page, M.: Analysis of genetic regulatory networks: a modelchecking approach. In: Benerecetti, M., Pecheur, C. (eds.) Working Notes of the Second Workshop on Model Checking and Artiﬁcial Intelligence, MoChArt 2003, Acapulco, Mexico, pp. 51–58 (2003) 7. Batt, G., Bergamini, D., de Jong, H., Garavel, H., Mateescu, R.: Model checking genetic regulatory networks using GNA and CADP. In: Graf, S., Mounier, L. (eds.) SPIN 2004. LNCS, vol. 2989, pp. 158–163. Springer, Heidelberg (2004) 8. McMillan, K.: Symbolic Model Checking. Kluwer Academic Publisher, Dordrecht (1993) 9. Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL: A Proof Assistant for HigherOrder Logic. LNCS, vol. 2283. Springer, Heidelberg (2002) 10. Bella, G.: Formal Correctness of Security Protocols. Information Security and Cryptography. Springer, Heidelberg (2007) 11. Gardner, T.S., Cantor, C.R., Collins, J.J.: Construction of a genetic toggle switch in Escherichia coli. Nature 403, 339–342 (2000) 12. Tian, T., Xu, S., Gao, J., Burrage, K.: Simulated maximum likelihood method for estimating kinetic rates in gene expression. Bioinformatics 23, 84–91 (2007)
Control Strategies for the Regulation of the Eukaryotic Heat Shock Response Elena Czeizler , Eugen Czeizler , RalphJohan Back, and Ion Petre ˚bo Akademi University Department of IT, A Turku 20520, Finland {elena.czeizler,eugen.czeizler,backrj,ipetre}@abo.fi
Abstract. Elevated temperatures cause proteins in living cells to misfold. They start forming larger and larger aggregates that can eventually lead to the cell’s death. The heat shock response is an evolutionary well conserved cellular response to massive protein misfolding and it is driven by the need to keep the level of misfolded proteins under control. We consider in this paper a recently proposed new molecular model for the heat shock response in eukaryotes, consisting of a temperatureinduced activation mechanism, chaperoning of misfolded proteins and selfregulation of the chaperon synthesis. We take in this paper a control driven approach to studying this regulatory network. We modularize the network by identifying its main functional modules. We distinguish three main feedback loops. The main question we are addressing is why is this level of complexity needed for implementing what could in principle also be achieved with an openloop design. We answer the question by comparing the numerical behavior of various knockdown mutants where one or more feedback loops are missing. We also discuss a new approach for a biologicallyunbiased model comparison.
1
Introduction
Decomposing large biological networks. Much experimental and theoretical eﬀort is invested nowadays in compiling large, systemlevel models for biochemical processes, including regulatory networks, signaling pathways, metabolic pathways, etc. Models can encompass many thousands of reactants and reactions, see [2]. On this scale, understanding the details of the network, especially the interactions among its various parts, or even noticing a highlevel functional separation in the network become considerable challenges. Recognizing that similar problems are also encountered in engineering (and elsewhere), see [3], one strategy towards a systemlevel understanding of a biological network is to adapt to systems biology speciﬁc methods coming from engineering sciences, in particular from control theory, see [6], [7], [10], [14], [15], [16], [17]. Distinguishing among the main functional modules of a biological network and identifying their individual contribution to the overall behavior can provide great insight into the basis of its reactivity and eﬃciency. Using a controldriven approach, one
Authors with equal contributions.
P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 111–125, 2009. c SpringerVerlag Berlin Heidelberg 2009
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often aims to identify the main regulatory components, including feedback and feedforward mechanisms. To disentangle their individual contribution to the network, knockdown mutants are often considered, see [13]. Such mutants, lacking one or more of the regulatory components, are then numerically compared in an eﬀort to identify and quantify the exact contribution of each component. A new approach for model comparison. Comparing alternative computational models for a biological process is in general a diﬃcult problem, where one has to consider the diﬀerences in the underlying reaction network, biological assumptions, kinetics, and initial conditions. When comparing alternatives that are submodels of a larger model, e.g., as in the case of a functional analysis of the various modules in a large network, the problem is somewhat simpler: the underlying reaction networks are very similar (albeit not identical), the biological constraints are the same (given by those of the reference model), and the kinetics of the reactions are the same (given by the reference model). The main question is that of how to choose the initial numerical setup of each model (the initial values of all variables in the models) in such a way that the comparison is unbiased and identiﬁes intrinsic diﬀerences in the structure of the models. One approach, related to a technique of mathematically controlled comparison, see [1], is to start from the initial values of the corresponding variables in the reference model, see [13] for a case study using this approach. While this approach is mathematically wellfounded, it may in fact lead to biased comparisons in the case of biological processes. Especially in the case of regulatory networks, models are assumed to be at steady state in the absence of the trigger of the response. Initial values are often chosen in such a way that the reference model is in steady state in the absence of the stimulus. However, imposing the steady state condition to the initial values of the reference model does not imply in general that a submodel will also be at steady state using the same initial values. As such, the behavior of the submodel will in fact exhibit the combined behavior of two diﬀerent eﬀorts: that of migrating from a possibly unstable state and, perhaps simultaneously, that of responding to a speciﬁc stimulus. Questions related to the speed of the response or its eﬀectiveness may thus get misleading answers. If the response to the stimulus of various diﬀerent submodels is to be compared, we argue in this paper that a diﬀerent approach yields biologically unbiased results. We propose an approach where the kinetics and the mass constants of the submodels are taken from the reference model. The initial distribution of reactants among their various forms is however diﬀerent from model to model and it is chosen in such a way that the initial setup of each model forms a steady state of that model in the absence of the stimulus. In particular, this approach allows to consider each of the submodels as a viable alternative model to the biochemical process of interest. Our case study: the heat shock response. We choose a recently proposed model for the eukaryotic heat shock response, see [11] and [12], as a case study for the model comparison method that we propose. The model is attractive for the purpose of this paper because on one hand, it is relatively small, with only
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ten reactants and twelve reactions, while on the other hand it contains a rather intricate control mechanism, with three diﬀerent feedback mechanisms. We analyze all eight knockdown mutants obtained by combining the three feedbacks and identify their individual contribution to four performance indicators of the reference model.
2
The Heat Shock Response in Eukaryotes
Molecular model. The main role in the response is played by the heat shock proteins (hsp). They act as chaperons for the misfolded proteins (mfp), by forming hsp: mfp complexes and assisting them to refold. In the model recently introduced in [12], the heat shock response is controlled by regulating the transactivation of the hspencoding genes. The transcription process for these genes can start only after some transcription factors, called heat shock factors (hsf), trimerize (hsf 3 ) and then bind to some particular DNA sequences, called heat shock elements (hse), which are the promoters of the hspencoding genes. The binding of a trimer hsf 3 to a heat shock element is denoted by hsf3 : hse. As an intermediary step before trimerization, the heat shock factors go ﬁrst through a dimerization stage, when they form hsf 2 complexes. Once the trimers hsf 3 are bound to the promoter sites, the transcription and translation of the hspencoding genes can start, leading eventually to the formation of new hsp molecules. When the level of hsp is high enough, the transcription process is turned oﬀ through a cunning selfregulating mechanism. The hsp proteins sequestrate the heat shock factors hsf in three ways: by binding to free hsf, by breaking dimers and trimers, as well as by unbinding hsf 3 from the DNA promoter sites and, at the same time, breaking the trimer hsf 3 . As a consequence, the heat shock factors are bound in hsp: hsf complexes and thus they are unable to form trimers anymore, turning oﬀ in this way the transcription of the hspencoding genes. As soon as the temperature is increased, proteins start misfolding, driving hsp away from hsf. Thus, hsf proteins are free and available to trimerize and bind to hse, promoting the synthesis of new hsp proteins. If the level of hsp becomes high enough, hsp unbinds hsf 3 from hse and again sequesters the majority of hsf into hsp: hsf, thus turning its own synthesis oﬀ. The molecular model for the heat shock response proposed in [12] is shown in Table 1. Model assumptions. For the sake of simplicity, the molecular model of [12] includes several reductions. For instance, even though in a cell there exist several types of slightly diﬀerent heat shock proteins, see [8], in this model they are all treated uniformly, with hsp 70 as base denominator. Similar conventiones are taken also for hse and hsf. The model does not diﬀerentiate either among diﬀerent other types of proteins present in a cell. From the point of view of the heat shock response, the only relevant feature is whether they are correctly folded, in which case they are gathered under the name prot, or misfolded, in which case they are called mfp. Some of the cellular mechanisms, such as protein synthesis and protein degradation, are also greatly simpliﬁed in the model. We refer to [12] for a detailed discussion on this issue.
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Table 1. The molecular model for the eukaryotic heat shock response proposed in [12] Reaction
Reaction name
2 hsf hsf 2
(hsf dimerization)
(1)
hsf + hsf 2 hsf 3
(hsf trimerization)
(2)
hsf 3 + hse hsf3 : hse
(DNA binding)
(3)
hsf3 : hse → hsf3 : hse + hsp
(hsp synthesis)
(4)
hsp + hsf hsp: hsf
(hsf sequestration)
(5)
hsp + hsf 2 → hsp: hsf + hsf
(hsf 2 breaking)
(6)
hsp + hsf 3 → hsp: hsf +2 hsf
(hsf 3 breaking)
(7)
hsp + hsf3 : hse → hsp: hsf +2 hsf + hse
(hspforced hsf 3 unbinding)
(8)
hsp → ∅
(hsp degradation)
prot → mfp
(protein misfolding)
hsp + mfp hsp: mfp
(protein chaperoning)
(11)
hsp: mfp → hsp + prot
(protein refolding)
(12)
(9) (10)
The model assumes three conservation relations: for the total amount of heat shock factors, for the total amount of proteins (except heat shock proteins and heat shock factors), and for the total amount of the heat shock elements: – [hsf] + 2 × [hsf 2 ] + 3 × [hsf 3 ] + 3 × [hsf3 : hse] + [hsp: hsf] = C1 , – [prot] + [mfp] + [hsp: mfp] = C2 , – [hse] + [hsf3 : hse] = C3 , for some constants C1 , C2 , C3 called mass constants. The only variable not covered by conservation relations is hsp, the main regulatory target of the network. Mathematical model. We associate to the molecular model in Table 1 a mathematical model in terms of diﬀerential equations as follows. A timedependant continuous variable is associated to each reactant and gives its concentration level. For a variable x, we denote by [x](t) its concentration at time t. The dynamics of the system is then described through a system of diﬀerential equations. Its formulation is based on the principle of mass action, see [4], [5]. For each variable, its diﬀerential equation gives the cumulated consumption and production rates of the corresponding reactant as an eﬀect of the reactions in Table 1. The rate of each reaction is, based on the principle of mass action, proportional with the concentration of the reactants. Due to lack of space we skip listing the resulting system of diﬀerential equations, referring instead to [11] and [12]. The kinetic rate constants and the initial values of all reactants were estimated in [12] so as to satisfy three conditions: (i) For a temperature of 37◦ C, the system is at steady state, i.e., the diﬀerentials of all variables are zero. This is a natural requirement since the model should exhibit no response in the absence of heat shock, i.e., at 37◦ C.
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(ii) For a temperature of 42◦ C, the numerical prediction of the model for [hsf3 : hse](t) should be in agreement with experimental data of [9] on DNA binding of hsf 3 . (iii) For a temperature of 42◦ C, the numerical prediction of the model for [hsp](t) should be correlated with experimental data of [12] on a denovo ﬂuorescent reporterbased experiment. The numerical setup of the model is in Table 2. We refer to [12] for details on parameter estimation and on model validation. The ﬁnal model exhibits the following major (numerical) achievements, inline with experimental evidence, see [12]: – (A) Makes economical use of the cellular resources: the hspencoding gene is only transactivated for a short while when exposed to heat shock. The gene transcription is virtually nonexistent in the absence of heat shock. – (B) It is fast to respond to a heat shock: the hspencoding gene is quickly transactivated in response to heat shock. – (C) The response is eﬀective: the mfp concentration is kept low for mild heat shocks. Table 2. The numerical values of the parameters and the initial values of the variables of the heat shock response model. A. The numerical values of the parameters. ki denotes the kinetic rate constant of the irreversible reaction (i). ki+ denotes the ‘lefttoright’ direction of reaction (i), while ki− denotes its ‘righttoleft’ direction. For the expression of the temperaturedependant parameter φT we refer to [11] and [12]. B. The initial values of all variables. A Param.
Value
k1+ k1− k2+ k2− k3+ k3− k4 k5+ k5− k6 k7 k8 k9 + k11 − k11 k12
3.49 0.19 1.07 10−9 0.17 1.21 · 10−6 8.3 · 10−3 9.74 3.56 2.33 4.31 · 10−5 2.73 · 10−7 3.2 · 10−5 3.32 · 10−3 4.44 13.94
Units ml #·s −1
s
ml #·s −1
s
ml #·s −1
s s−1 ml #·s −1
s
ml #·s ml #·s ml #·s −1
s
ml #·s −1
s s−1
B Variable
Initial conc.
[hsf] [hsf 2 ] [hsf 3 ] [hse] [hsf3 : hse] [hsp] [hsp: hsf] [mfp] [hsp: mfp] [prot]
0.67 8.7 · 10−4 1.2 · 10−4 29.73 2.96 766.88 1403.13 517.352 71.65 1.15 × 108
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– (D) Scalable: higher response for higher temperature. The transactivation of the hspencoding gene reaches a higher peak and/or remains longer on the 100% level under higher temperature. We disentangle in this paper the contribution of the various modules in the network to achieving these properties. We discuss the methodology in Section 3 and apply the method to the analysis of the heat shock response model in Section 4.
3
A Functional Decomposition of the Heat Shock Response Model
A controldriven approach. When designing the modular decomposition of a process, the ﬁrst step is to identify and separate the process to be regulated, called the plant. The current state of that process is captured by speciﬁc sensors which send the information to a decisionmaking module, called the controller. The decision taken by the controller is implemented through a device, called the actuator, that modiﬁes the state of the process, thus inﬂuencing the activity of the plant. A key concept of control systems is the existence of some feedback mechanisms, which are used to cope with the uncertainties of the system. A feedback sensor sends the current state of the process back to the controller, thus facilitating a dynamical compensation for disturbances of the system. In complex applications, the plant (and by eﬀect, the controller, the sensors, and the actuators) can be chosen in several diﬀerent ways depending on what is deemed to be the main target of the system, thus leading to several possibly diﬀerent decompositions of the system. An easy example illustrating these concepts and their interactions is given by the functioning principles of a motion activated spotlight. In this case, an electronic control unit, i.e., the controller of the system, reads the signal from the motion sensor in order to determine if there is any change in the environment. If the sensor’s input falls outside of the parameters of the system’s settings, then the control unit triggers a relay switch, i.e., the actuator, that operates the lighting system. As long as the sensor detects movement, it feeds this information to the control unit, keeping the switch on. Modularization of the heat shock response model. In the case of the eukaryotic heat shock response we set the plant to focus on the main target, i.e., the misfolding and refolding of proteins. As protein refolding can be done only with the help of the chaperons hsp, we assign the actuator to model their synthesis and degradation. The controller’s task is set to modulate the level of hsf3 : hse, since the DNA binding level determines the chaperon production. The controller’s activity is inﬂuenced also by the information received from a sensor which measures the level of chaperons in the system. When this is high enough, the chaperons will sequestrate the heat shock factors by forming complexes hsp: hsf, leading in this way to a decrease in the level of hsf3 : hse. In particular, depending on the type of reactions which lead to the formation of the complex hsp: hsf, we identify three feedback mechanisms:
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Table 3. The functional decomposition of the model in Table 1. We denote the ‘lefttoright’ direction of reaction (5) by (5)+ and by (5)− its ‘righttoleft’ direction. Plant Actuator Sensor Controller Feedback F B1 Feedback F B2 Feedback F B3
Main Task Protein misfolding and refolding Regulate the level of hsp Measure [hsp] Modulate the level of DNA binding Sequestration of hsf Dimer and trimer breaking hspforced DNA unbinding
Reactions (10), (11), (12) (4), (9) (1), (2), (3), (5)− (5)+ (6), (7) (8)
Fig. 1. The control structure of the heat shock response network +
– F B1 : sequestration of hsf, i.e., reaction (5) (i.e., the ‘lefttoright’ direction of reaction (5)); – F B2 : breaking hsf dimers and trimers, i.e., reactions (6) and (7); – F B3 : unbinding hsf 3 from hse, i.e., reaction (8). The functional decomposition of our model is summarized in Table 3, where the reaction numbers refer to the reactions in Table 1. A graphical illustration of the decomposition is in Fig. 1.
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Disentangling the Functional Roles of the Control Feedbacks
Approach. We consider the eight knockdown mutants obtained through all combinations of feedbacks F B1 , F B2 , and F B3 . We analyze the dynamical behavior of the mutants at 42◦ C. We choose this particular temperature since at 42◦ C the experimental data shows both a pronounced heat shock response in terms of increased levels of misfolded proteins, as well as an explicit response in terms of increased, transient DNA binding of hsf 3 , see [12]. Unbiased model comparison. In order for our analysis to be biologically unbiased, we aim to eliminate all accidental diﬀerences between our models. In particular, they should satisfy some initial biological constraints. This way, we
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make sure that (i) the remaining dissimilarities in their dynamical behaviors reﬂect intrinsic diﬀerences in their architecture and (ii) they are all fairly considered as viable alternative models for the heat shock response. We impose the following two constraints. (i) The reactions of all knockdown mutants assume kinetic rate constants identical to those of the corresponding reactions in the reference model. Also, the values of the mass constants C1 , C2 , C3 of each of the mutants are identical to those of the reference model. Both these conditions are natural since each knockdown mutant is a submodel of the reference model. As such, they should all assume identical chemistry and mass constants. (ii) The initial values of the variables of each knockdown mutant form a steady state of that model for a temperature of 37◦ C. Note that the same condition has been applied in [12] when choosing the initial values of all variables for the reference model. Note that our constraint (ii) is fundamentally diﬀerent than that of [13]. Rather than taking an approach based on mathematically controlled comparison, see [1], where the submodels start from the same initial setup, we take a diﬀerent view based on biologically meaningful constraints. We consider each submodel as a viable alternative model for the heat shock response and as such, we assume that before the heat shock is applied, they are in a steady state (at 37◦ C). Consequently, we impose the condition that the initial values of each submodel form a steady state of that model at 37◦ C. In particular, each model assumes a diﬀerent initial numerical setup, determined by the kinetics of its underlying reaction network and by the mass constants. Note also that we do not perform parameter estimation for each of the mutants. Indeed, this approach would yield diﬀerent kinetics for each model and thus, the contribution of each feedback to the network would be wildly diﬀerent from mutant to mutant. Clearly, the results of our analysis remain heavily dependent on the numerical values of the mass constants and of the kinetic rate constants chosen in [12] for the reference model. We discuss this dependency in Section 5. The knockdown mutants. Each of the knockdown mutant models will be denoted as MX where X ⊂ {1, 2, 3} represents the indexes of the feedback mechanisms contained in MX , as follows: i) M0 consists of reactions (1)(4), (9)(12). Using controltheory terminology, this model is referred as the openloop design. ii) M1 consists of reactions (1)(5), (9)(12). iii) M2 consists of reactions (1)(4), (6)(7), (9)(12). iv) M3 consists of reactions (1)(4), (8)(12). v) M1,2 consists of reactions (1)(7), (9)(12). vi) M1,3 consists of reactions (1)(5), (8)(12). vii) M2,3 consists of reactions (1)(4), (6)(12).
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Fig. 2. The total amount of mfp in the seven knockdown mutants and in the reference model for a constant temperature of 42◦ C
Numerical comparison of the mutants. The main task of the heat shock response is to keep the level of misfolded proteins under control. Focusing on this aspect alone, all knockdown mutants (even the openloop design M0 ), excel in achieving the goal. Perhaps surprisingly, they actually perform much better from this point of view than the reference model M1,2,3 , see Fig. 2: the total amount of mfp at a temperature of 42◦ C is considerably smaller in all mutants than in the reference model. On the other hand, it is essential for the cell to be eﬃcient in the use of materials and energy. In the case of the heat shock response, this translates in the amount of chaperons needed in each of the models to achieve the response. From this point of view, the tradeoﬀ in the mutants is very high. Indeed, all mutants maintain much higher amounts of chaperons throughout the response than the reference model. For a comparison in relative terms we associate a timedependant cost function describing the eﬀort in each model with respect to its result in terms of chaperons per unit of misfolded protein: cost(t) = [total HSP] / [total MFP] = ([hsp](t) + [hsp: hsf](t) + [hsp: mfp](t))/([mfp](t) + [hsp: mfp](t)). The evolution of the cost function for all models is illustrated in Fig. 3 and shows that all mutants are overreacting, by overproducing hsp in an eﬀort to maintain a very low level of mfp. (A similar plot can be obtained also by comparing the level of chaperons in the models in absolute terms.) The overreaction to heat shock is also clearly seen in terms of DNA binding and gene transcription, see Fig. 4: DNA binding remains high throughout the heat shock response in all mutants, while it is only transient in the reference model.
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Fig. 3. The amount of chaperons per unit of misfolded protein in the seven knockdown mutants and in the reference model for a constant temperature of 42◦ C
The contribution of the three feedbacks. We analyze in this section the functionality of each of the three feedbacks from the point of view of their contributions to the properties (A)(D) of the heat shock response model, see Section 2. For this, we compare the eight knockdown mutants to capture the eﬀects of adding or removing each of the feedbacks. In line with biologically unbiased comparison discussed in this paper, we set the initial values for each of the knockdown mutants to obtain a steady state of the system for a temperature of 37◦ C. As such, a large part of the diﬀerences among the various mutants can already be observed from their initial values, i.e., from their behavior in the absence of the heat shock. The initial numerical setup of all models is given in Table 4. The mutants M0 , M1 , M2 , M3 , and M2,3 exhibit DNA binding at the 100% level as their initial setup (equivalently, the initial value of [hse] is negligible in Table 4. The initial numerical setup of all knockdown mutants. The initial values are chosen so that each mutant is at steady state for a temperature of 37◦ C. The initial value of [prot] in all models is identical to its value in the reference model, see Table 2B. M0 M1 M3 M2 M2,3 M1,2 M1,3 M1,2,3
hsf 0.0028 0.0028 0.2018 35.168 35.1759 0.1316 0.1172 0.6692
hsf 2 0.0001 0.0001 0.3516 0.2178 0.218 0 0.1525 0.0009
hsf 3 438.2 416.803 437.893 22.3266 22.1625 0 0.0069 0.0001
hsf3 : hse 32.6895 32.6895 32.6884 32.6895 32.6693 14.6625 16.405 2.9565
hse 0 0 0.001 0 0.0202 18.027 16.2845 29.733
hsp 8479.29 8479.29 8479.03 8479.29 8474.06 3803.28 4255.28 766.875
hsp: hsf 0 64.1905 0.0212 1212.02 1212.56 1368.55 1363.01 1403.13
hsp: mfp 71.6476 71.6476 71.6476 71.6476 71.6476 71.6476 71.6476 71.6473
mfp 46.79 46.79 46.7914 46.79 46.8189 104.317 93.2361 517.352
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Fig. 4. The level of DNA binding in the seven knockdown mutants and in the reference model. The graph plots [hsf3 : hse](t) relative to the total number of heat shock elements (identical in all models) for a constant temperature of 42◦ C.
all of these mutants). Thus, in these cases, the hspencoding gene is transactivated at the 100% level even in the absence of heat shock. This is not only a contradiction of experimental data of [8], but also a noneconomical use of the cellular resources, see also Fig. 3 for a measure of the cost of the response. These models fail by overreaching their target: the total amount of mfp in the system is kept at an unnecessarily low level, see Fig. 2. Indeed, the reference model allows for a higher [mfp] (that is still negligible relative to [prot]), for a dramatically lower price in terms of [hsp]. As such, we conclude already from their predicted relentless DNA transcription activity under physiological conditions that these mutants are nonviable models for the heat shock response. In terms of the amount of chaperons in the system under a constant heat shock at 42◦ C, F B1 appears to be the main driver to minimize [hsp]. Indeed, removing F B1 from the reference model yields model M2,3 , which is the only mutant containing two feedbacks that exhibits a behavior even worse than that of the openloop model M0 (that has no feedbacks at all), see Figs. 24. Regarding the role of F B1 , we conclude that no viable heat shock response can be obtained in the absence of feedback F B1 . At the same time, F B1 alone (see model M1 ) cannot lead to a viable response either. Regarding F B2 , we note that in models M2 and M2,3 , the level of hsf 3 at 37◦ C is one degree of magnitude smaller than in the models M0 , M1 , and M3 . Even though the kinetic details of the reactions do not propagate this change towards a lower level of hsf3 : hse, this observation points to a critical role of F B2 in lowering the level of hsf3 : hse. The same eﬀect cannot be obtained by means of either F B1 alone, or F B3 alone. Regarding the role of F B3 , consider now adding F B1 to either of the other two feedbacks. We obtain models M1,2 and M1,3 that have drastically diﬀerent behaviors under physiological conditions than the other knockdown mutants: the DNA binding level, [hsf3 : hse], is lowered from the 100% level in the absence
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of F B1 to around the 50% level, see Fig. 4. As a direct consequence, [hsp] is lowered by about 50% and, as a tradeoﬀ, [mfp] increases by about 50%. There is no big numerical diﬀerence between the two models at 37◦ C. When a heat shock of 42◦ C is applied however, the behavior of the two models diﬀers, especially in the speed of the response. While the response of M1,2 in terms of DNA binding is static ([hsf3 : hse] remains roughly constant throughout the response, see Fig. 4), that of M1,3 is adaptive. It is in fact the only knockdown mutant exhibiting an adaptive behavior in terms of DNA binding. Similarly as in the case of the reference model, [hsf3 : hse] is transiently increased in M1,3 during the heat shock and eventually returns to the basal levels. Moreover, both the time to reach the peak of the response, as well as the time to return to the physiological level of [hsf3 : hse] in M1,3 , are very similar to those in the reference model. We conclude then that F B3 (in addition to F B1 ) has a main role in achieving a fast response to heat shock. Indeed, this conclusion is also supported by the plots in Fig. 2, showing that [mfp] is lowered very slowly in the absence of F B3 . Moreover, as seen in Fig. 4, in the absence of F B3 the DNA binding levels are insensitive to a temperature upshift to 42◦ C. For a heat shock at 43◦ C, the only mutant exhibiting a diﬀerent response than at 42◦ C is M1,3 . Similarly to the reference model, [hsf3 : hse] reaches a higher peak in M1,3 , while it remains essentially unchanged for the other mutants. Based on the numerical observations above, we can now summarize the contribution of the three feedbacks to the performance indicators (A)(D) of the network as follows: – (A) Makes economical use of the cellular resources (the hspencoding gene is only transactivated for a short while when exposed to heat shock): F B1 plays the major role here, F B2 , F B3 are also important. In the absence of F B1 , gene transcription is at the 100% level even without heat shock. – (B) It is fast to respond to a heat shock (the hspencoding gene is quickly transactivated in response to heat shock): F B3 , F B1 play the major role here. Model M1,3 is as fast to react to heat shock as the reference model, albeit the scope of its response is lower than that of the reference model. – (C) The response is eﬀective (the mfp concentration is kept low for mild heat shocks): the underlying openloop structure is enough to achieve this. No feedback mechanism is needed for maintaining a low [mfp] for a heat shock at 42◦ C. Rather, the role of the feedbacks is to minimize the cost of the response. – (D) Scalable (higher response for higher temperature): F B3 , F B1 play the major role here. M1,3 is the only mutant that scales its response to higher temperatures, similarly as the reference model. For higher temperatures, gene transactivation is faster to raise and it raises to a higher level. In the list above, the role of F B2 as a sole or a main contributor to one of the properties (A)(D) is not clearly distinguished. F B2 appears more as a modulator for F B1 and F B3 . There is a clear numerical argument for this milder eﬀect: F B2 consists of a regulation mechanism acting on tow reactants, hsf 2 and hsf 3 , having
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negligible concentrations in the reference model both at 37◦ C and throughout the response at 42◦ C. In this context, it is most surprising however to see the dramatic eﬀect that the removal of F2 has on the reference model, see, e.g., Fig. 4.
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Numerical model comparison is a diﬃcult problem even in the case of submodels of a larger network. We argued in this paper that the common practice of assuming for all submodels the same initial setup as that of the reference model is biologically biased. Indeed, a numerical setup chosen so that the reference model satisﬁes some biological constraints will not necessarily ensure the same for its submodels. As such, the submodels will already from the start be deemed as unviable alternative models for the biological process under study. We considered a diﬀerent approach where the kinetics and the mass constants are taken from the reference model (so that the chemistry and the total amount of reactants are the same in all submodels), but the initial setup of each submodel is chosen in such a way that: (i) it satisﬁes all mass conservation relations and (ii) it satisﬁes all biological assumptions. We argued that in this way we obtain a biologically unbiased model comparison, where each submodel is evaluated as a real alternative model for the biological process. Extending the idea of biologically unbiased model comparison to models that assume diﬀerent underlying reaction networks is appealing. One possible way to do it is by combining numerical approaches such as those demonstrated in this paper with qualitative approaches of model reﬁnement such as those developed in software engineering. Two diﬀerent models to be compared could ﬁrst be embedded into a larger model in such a way that they both become its submodels. Questions related to a systematic methodology for model embedding, dealing with conﬂicting model assumptions, and other issues appear to be very interesting. Similarly to many other numerical techniques such as sensitivity analysis, the numerical comparisons demonstrated in this paper depend heavily on the numerical setup of the models, i.e., on the numerical values of the kinetic rate constants, mass constants, and on the initial values of the variables. We say then that our analysis is local. When repeating the analysis in a diﬀerent numerical setup, the conclusions regarding the role of each of the feedbacks could be very diﬀerent. This is most clear if we consider changes in the kinetic rate constants: the control in the regulatory network can easily be shifted elsewhere by drastically slowing down some reactions and speeding up others. The same is true however also when changing the mass constants. For example, we considered an experiment where we increased the total amount of hsf from around 1412 as in this paper to about 10000. We then changed all initial values of all variables in each of the models so that they form a steady state of the model for a temperature of 37◦ C. When repeating the knockdown analysis in this setup, the conclusions were very diﬀerent. It turned out that as far as the cost function is concerned, the openloop design performs almost as well as the reference model, while the mutants
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consisting of two feedbacks perform much worse. As far as gene transcription activity is concerned under a constant heat shock at 42◦ C, the reference model exhibits a similar shape as in Fig. 4, while the mutants show a constant activity at the 100% level. Taking a similar approach in the case when the total amount of hsf is lowered instead to around 500, we obtained that the reference model has by far the lowest cost, with M1,2 and M1,3 performing well, and the openloop model much worse. The gene transcription activity is similar to those in Fig. 4, except for M1,2 which shows a lower transactivation level. Two conclusions can thus be formulated. On one hand, this is a local numerical analysis, in the same category with approaches such as parameter ﬁt and sensitivity analysis. They all have to be taken in close relationship with the experimental data and available biological knowledge and validated as such. On the other hand, repeating the analysis in diﬀerent, albeit less validated, numerical contexts can be very useful. For example, modules whose functional role is hidden by other, more dominating modules, can be easier to analyze in numerical contexts where they assume the dominant role. Clearly, projecting conclusions from one numerical context to another is a challenge in itself that has to be addressed.
References 1. Alves, R., Savageau, M.A.: Extending the method of mathematically controlled comparison to include numerical comparisons. Bioinformatics 16, 786–798 (2000) 2. Chen, W.W., Schoeberl, B., Jasper, P.J., Niepel, M., Nielsen, U.B., Lauﬀenburger, D.A., Sorger, P.K.: Input output behavior of ErbB signaling pathways as revealed by a mass action model trained against dynamic data. Molecular Systems Biology 5, 239 3. Csete, M.E., Doyle, J.C.: Reverse Engineering of Biological Complexity. Science 295, 1664–1669 (2002) 4. Guldberg, C.M., Waage, P.: Studies Concerning Aﬃnity. C. M. Forhandlinger: VidenskabsSelskabet i Christiana 35 (1864) 5. Guldberg, C.M., Waage, P.: Concerning Chemical Aﬃnity. Erdmann’s Journal f¨ ur Practische Chemie 127, 69–114 (1879) 6. Hawkins, B.A., Cornell, H.V. (eds.): Theoretical Approaches to Biological Control. Cambridge University Press, Cambridge (1999) 7. Kitano, H.: Systems biology: A brief overview. Science 295, 1662–1664 (2002) 8. Holmberg, C.I., Tran, S.E., Eriksson, J.E., Sistonen, L.: Multisite phosphorylation provides sophisticated regulation of transcription factors. Trends Biochem. Sci. 27(12), 619–627 (2002) 9. Kline, M.P., Morimoto, R.I.: Repression of the heat shock factor 1 transcriptional activation domain is modulated by constitutive phosphorylation. Molecular and Cellular Biology 17(4), 2107–2115 (1997) 10. Lazebnik, Y.: Can a biologist ﬁx a radio?  or what I learned while studying apoptosis. Cancer Cell 2, 179–182 (2002) 11. Petre, I., Mizera, A., Hyder, C.L., Mikhailov, A., Eriksson, J.E., Sistonen, L., Back, R.J.: A new mathematical model for the heat shock response. In: Kok, J. (ed.) Algorithmic bioprocesses. Natural Computing. Springer, Heidelberg (2008) 12. Petre, I., Hyder, C.L., Mizera, A., Mikhailov, A., Eriksson, J.E., Sistonen, L., Back, R.J.: A simple mathematical model for the eukaryotic heat shock response (submitted, 2009)
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Computing Reachable States for Nonlinear Biological Models Thao Dang, Colas Le Guernic, and Oded Maler cnrsverimag, 2, av. de Vignate, 38610 Gieres, France
Abstract. In this paper we describe reachability computation for continuous and hybrid systems and its potential contribution to the process of building and debugging biological models. We then develop a novel algorithm for computing reachable states for nonlinear systems and report experimental results obtained using a prototype implementation. We believe these results constitute a promising contribution to the analysis of complex models of biological systems.
1
Introduction
The development of modeling formalisms and analysis techniques for the study of biological systems is a central topic in systems biology. The formalisms proposed for representing biological processes are very diverse, diﬀering at the levels of abstraction, time scales and types of dynamics. The formalism chosen depends naturally on the level of detail needed to answer the speciﬁc biological question and on the granularity of available experiments. The contribution of this work is at the level of abstraction of ordinary diﬀerential equations (ODEs), a widely used modeling formalism. Biological systems, for instance metabolic networks consisting of sets of reactions, can be viewed as continuous dynamical systems with state variables denoting concentrations. The resulting diﬀerential equations are derived, for example, from mass action rules and are, typically, polynomial. Such equations can be numerically simulated from a given initial condition provided that the exact values of the parameters and the external environmental conditions are known. In certain restricted cases it is possible to determine global properties analytically. Though widely used, ODEs suﬀer from several limitations. First, the passage from a ﬁnite number of molecules to realvalued concentration is not always justiﬁed, especially when the number of molecules is small [22]. Secondly, many biological phenomena, for example gene activation, are more naturally modeled as transitions between discrete states. Pure ODEs cannot easily accommodate this mixture of continuous evolutions and discrete events. Alternatively, purely discrete formalisms, based on transition systems expressed in various syntactic forms, suﬀer from a similar reciprocal limitation in the sense of not being amenable to quantitative reasoning.
Part of this work was done while the third author was a Weston visiting professor at Weizmann Institute.
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Second, the lack of quantitative information concerning molecular concentrations, reaction rates and other parameters is the rule, not the exception, in Biology. Consequently the value of predictions obtained using numerical ODEs models, where the values of the parameters are “guessed” or “tuned”, is severely limited. Moreover, the validation of models based on ODEs with poorlyknown parameters is diﬃcult if not impossible because we are never sure to have covered all the qualitative behaviors compatible with a model by performing only a ﬁnite number of simulations, each with a diﬀerent choice of parameters. This fact limits the applicability of such models for testing biological hypotheses. To deal with this problem, qualitative approaches, notably based on qualitative versions of diﬀerential equations, have been proposed for representing genetic regulatory networks, molecular interaction networks or metabolic pathways [30, 40]. In these models only the direction of inﬂuence between variables is encoded (e.g. activation vs. inhibition) and much of the quantitative information is absent. As a consequence of such underconstrained descriptions, purelyqualitative approaches often lead to overlyconservative results in the sense of admitting many spurious behaviors. We propose a technique that can be used to analyze in a systematic manner quantitative models admitting this kind of uncertainty whose nature is settheoretic rather than stochastic. The analysis techniques that we use and extend originate from the study of hybrid dynamical systems, a domain situated in the intersection of control theory and computer science and are based on reachability analysis of hybrid automata. As their name suggests, hybrid automata are the result of marrying automata with diﬀerential equations. Each discrete state (mode) of the automaton is associated with one set of diﬀerential equations according to which the continuous variables evolve while being in that mode. When the variables satisfy certain conditions (transition guards) the automaton may switch to another mode where another set of equations will govern the evolution of the continuous variables. While hybrid automata allow us to express piecewisecontinuous processes and can underlie numerical simulation, much of the analytic reasoning available for purelycontinuous systems (especially for linear ones) is lost due to switching. In the last couple of years new techniques have been developed for the algorithmic analysis of hybrid systems, which open as well new opportunities for the analysis of purelycontinuous systems subject to uncertainties. These techniques combine ideas from control theory, numerical analysis, graph algorithms and computational geometry in order to export algorithmic veriﬁcation, also known as model checking, to the continuous and hybrid domains. The principles of algorithmic veriﬁcation can be summarized as follows. The system in question is modeled as an automaton whose transitions are labeled by input events. These inputs represent interactions of the automaton with its external environment (users, other systems). Each sequence of input events induces one behavior of the automaton, a trajectory over its state space. Simulation is the process of stimulating the automaton progressively with one input sequence and observing the behavior that this sequence induces starting from a given initial state. The problem is that the number of such sequences is prohibitively large.
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Veriﬁcation is based, instead, on computing with sets of states: starting from an initial set of states P0 , one computes all the onestep successors of P0 (under all possible inputs) to obtain the set P1 , to which the same procedure is applied until all the states reachable from P0 under any admissible input are computed.1 Showing, for example, that some “bad” set of states is never reached (a “safety” property) amounts to checking whether the reachable set thus computed intersects the bad set. This computation replaces an inﬁnite (or just huge) number of simulations. More complex properties that specify some temporal patterns of events can be speciﬁed and veriﬁed as well using similar methods. The adaptation of this idea to continuous systems works as follows. Consider a diﬀerential equation of the form x˙ = f (x, v) where x is a vector of state variables and v represents external disturbances and parameter uncertainties which are not known exactly but are always taken from a bounded convex set V . Given a subset P0 of the state space (in a form of, say, a polytope) and a time step r, one can compute another polytope P1 , which contains all the points reachable from P0 within the time interval [0, r] under any admissible value of v during that interval. Repeating this process we can obtain an over approximation of all the reachable states in any desired time horizon. To give a concrete example, one can compute all the possible evolutions of a reaction under all possible concentrations of a signalling molecule which are typically not precisely known, but which remain in a known interval. The principal contribution of this paper is in developing a new technique for conducting this type of analysis for nonlinear systems and in demonstrating its applicability on several biological models. The rest of the paper is organized as follows. In Section 2 we give a brief introduction to the stateoftheart in reachability computation for linear systems and explain why it cannot be applied in a straightforward manner to nonlinear systems. We then describe the hybridization approach [6] for handling nonlinear systems. Hybridization is based on over approximating a nonlinear system by a piecewiseaﬃne system, a restricted type of a hybrid automaton without discontinuous jumps. Although, in principle, hybridization provides for the application of linear techniques to nonlinear systems, it suﬀers from inherent limitations that restrict its applicability to very lowdimensional systems. Section 3 describes our major contribution, a new dynamic hybridization scheme in which linearization is not based on a ﬁxed partition of the state space and thus avoids much of the associated state explosion. For this algorithm we provide in Section 4 compelling experimental results, analyzing highlynonlinear systems of 6 and 9 variables taken from systems biology. We conclude with a discussion of future work. Although we have tried to maintain the paper as self contained as possible, some readers might want to consult books like [42, 39, 28, 38] for some notions of geometry, linear algebra and dynamical systems or expository articles such as [34, 35] which discuss similarities and diﬀerences between transition systems and continuous dynamical systems. 1
More precisely, the computation is guaranteed to converge for ﬁnitestate systems. In continuous domains we are currently satisﬁed with a bounded time horizon [34].
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Reachability: Linear and Nonlinear Systems
Computing the states reachable by all trajectories of a dynamical system subject to disturbances and parameter variations emerged as a new research topic from the interaction between computer science and control. Reachability computation can be seen as a peculiar way to conduct exhaustive simulation which can be useful for the analysis of control systems, the veriﬁcation of analog circuits, the debugging of biological models and, in fact, any other activity based on dynamical systems models. After a decade of intensive research, [2,25,11,15,26,4, 33,37,10,5,12,32,24] it is fair to say that a satisfactory solution has been provided for timeinvariant linear systems. Existing algorithms manage to produce, within seconds, highquality approximations of the reachable states of linear systems with hundreds of state variables, for time horizons of thousands of integration steps. Notwithstanding these achievements, the real challenge in almost any application domain, Biology included, is the treatment of nonlinear systems, a challenge that we address in the present paper. Let us recall the rules of the game. Given a dynamical system S deﬁned by a diﬀerential equation x˙ = f (x, v) with v ranging over some bounded set V , a set P of initial states and some time horizon h, we would like to compute the set of states reachable from points in P by trajectories of S within some t ∈ [0, h]. Fixing some time discretization step r, the reachable set is approximated by the union of the sets in a sequence P0 , P1 , . . . where P0 contains all states reachable from P within t ∈ [0, r] and each Pi+1 includes states reachable from Pi within r time. Actual computations often work ﬁrst in discrete time where Pi+1 is reachable from Pi in one time step and then some error terms are added to bloat Pi+1 and compensate with respect to continuous time. Reachability computation of linear systems is relatively easy. Consider ﬁrst a discretetime autonomous linear system deﬁned by x = Ax and a set P which admits a ﬁnite representation, for example, a polytope represented by its vertices or supporting halfspaces, an ellipsoid represented by its center and deformation matrix or a zonotope represented by its center and generators. Then the linear transformation “commutes” with the representation. For example, if P = conv(P˜ ), meaning a polytope P being the convex hull of its ﬁnite set of vertices P˜ , then A · P = A · conv(P˜ ) = conv(A · P˜ ), (1) that is, the vertices of the polytope obtained by applying A to the whole set P are the result of applying A to the vertices of P . The extension of this idea to systems with underspeciﬁed input, that is, x = Ax + v where v ranges over a bounded convex set V , is more involved. The set of onestep successors of a set P under such a dynamics is captured by the Minkowski sum P = AP ⊕ V , which yields a polytope P with more vertices than P . This repeated growth in the size of the representation of Pi makes it impractical to iterate for a long time horizon because the number of points on which A has to be evaluated becomes huge. Two approaches are commonly used to alleviate this problem:
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1. For ellipsoids or for polytopes represented by their supporting halfspaces one can use techniques based on the maximum principle [41,13] to obtain an over approximation of AP ⊕ V whose representation size is not much larger than that of P ; 2. The modiﬁed recurrence scheme of [27, 24] keeps the number of points to which the linear transformation is applied ﬁxed. Its implementation using zonotopes [23,24], a subclass of polytopes which are closed under Minkowski sum, provides a very eﬃcient solution which is, practically, exact for discrete time. The technique that we present in this paper is invariant under the choice between these two approaches so we express it in terms of an abstract successor operator σ which, given a set P , an aﬃne diﬀerential inclusion (see below) of the form x˙ ∈ Ax ⊕ V and a time step r, it produces the set σ(P, A, V, r) containing all points reachable from points in P by trajectories of duration r of the aﬃne dynamics. The generic linear reachability algorithm can then be written as: Algorithm 1 (Linear Reachability) ˜ [0,r] (P ) P0 := R repeat i = 1, 2, . . . Pi := σ(Pi−1 , A, V, r) until i = k ˜ [0,r] (P ), the over approximation of the states reachable from P within The set R the time interval [0, r], can be computed, for example, by bloating the convex hull of P ∪ σ(P, A, V, r) as in [4] or [6]. Moving to nonlinear systems of the form x = f (x) for arbitrary f one observes that “convexity” properties such as (1) do not hold and new ideas are needed. In principle, it is possible to evaluate f on some representative ﬁnite sample P˜ ⊂ P and then use the resulting points to construct a set which over approximates f (P ). However, the approximation can be very coarse and will require a costly optimization procedure to be reﬁned, something that cannot be aﬀorded as part of the inner loop of the reachability algorithm. The “hybridization” technique of [6] suggests a good tunable compromise between the quality of the approximation, the diﬃculty of the computation and the frequency in which it is invoked. Before explaining the idea, let us give some necessary deﬁnitions. We consider a state space X, a bounded subset of Rn equipped with a metric ρ. Given two bounded closed subsets Y and Y of X, the Hausdorﬀ distance between them (the lifting of ρ to sets) is ρ(Y, Y ) = max{max min ρ(y, y ), max min ρ(y, y )}. y∈Y y ∈Y
y ∈Y y∈Y
The trajectories of a dynamical system are viewed as signals over X. Definition 1 (Signals). A signal over X is a partial continuous function ξ from T = [0, ∞) to X whose domain of deﬁnition is T or a preﬁx [0, r] of it. In the latter case we say that ξ is ﬁnite with duration r. The concatenation of a
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ﬁnite signal ξ deﬁned over [0, r] and a signal ξ satisfying ξ (0) = ξ(r) is deﬁned in the obvious way and is denoted by ξ · ξ . The continuous equivalent of a nondeterministic automaton is the relational vector ﬁeld, also known as diﬀerential inclusion [7]. Definition 2 (Relational Vector Fields). A relational vector ﬁeld over X is a function f : X → 2X − {∅} which is assumed to be KLipschitz, satisfying ρ({x}, {x }) < a ⇒ ρ(f (x), f (x )) < Ka. When f is a (deterministic) function we write f (x) = y rather than f (x) = {y}. Definition 3 (Dynamical Systems, Trajectories and Reachable Sets). A (continuous) dynamical system is a pair S = (X, f ) where X is a state space and f is a vector ﬁeld. A trajectory of S starting from x is a signal ξ over X with ξ(0) = x and for every t in the domain of deﬁnition of ξ, ξ(t) ∈ X and dξ(t)/dt ∈ f (ξ(t)). The set of all trajectories of S starting from any x ∈ P is denoted by L(S, P ). The sets of states reachable from P within a time interval [h, h ] is R[h,h ] (P ) = {ξ(t) : ξ ∈ L(S, P ) ∧ t ∈ [h, h ]}. Hybridization takes a nonlinear system S = (X, f ) and produces another dynamical system (S , f ) which over approximates it, that is, L(S, P ) ⊆ L(S , P ) for every P , and then computes the reachable states of S . A formal deﬁnition of S as a hybrid automaton can be found in [6]. Since our algorithm does not use hybrid automata explicitly we only give an informal explanation. Consider a partition of X into hyper rectangles (we use the term box hereafter). For each box Xq one can compute a linear function Aq and an error polytope Vq such that for every x ∈ Xq , f (x) ∈ Aq x ⊕ Vq . In other words, Aq is a local linearization of f with error bounded in Vq . Thus the vector ﬁeld f is deﬁned as f (x) = Aq x ⊕ Vq iﬀ x ∈ Xq . To perform reachability computation on S one applies linear reachability using Aq and Vq as long as the reachable states remain within box Xq . Whenever some Pi reaches the boundary between Xq and Xq it is intersected with the switching surface (the transition guard, in the terminology of hybrid automata) and the obtained result is used as an initial set for reachability computation in q using Aq and Vq , as illustrated in Figure 1(a,b). The main advantage of hybridization is that the costly procedure of ﬁnding a good linear approximation is not invoked in every step, only in the passage between boxes. Although this scheme is clean and general, it suﬀers from some serious diﬃculties on the way to realization: – Although the intersection of the actual set of reachable states inside a box with a facet of the box is typically a convex set, its computation can be inefﬁcient and inaccurate. To see why, consider a subsequence of sets Pj , . . . , Pk computed using some linear technique, all of which intersect the boundary
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A1
A2
(a)
(b)
Fig. 1. Computing reachable states of the hybridization: (a) applying linear reachability using A1 until intersection with the boundary; (b) taking the intersection as an initial set for linear reachability using A2
(a)
(b)
(c)
Fig. 2. (a) the intersection with the boundary spans over several iterations; (b) continuing with each intersection separately; (c) continuing with an approximation of the union of intersections
G as illustrated in Figure 2(a). In this case we have either to spawn several computations with the dynamics of the subsequent box, each starting with some Pi ∩ G (Figure 2(b)) or to over approximate i Pi ∩ G by a convex set, an operation that may lead to a large overapproximation error (Figure 2(c)). – The size of the partition of the state space is, of course, exponential in the dimension, hence care should be taken in order to avoid state explosion. As suggested in [6], the partition can be generated ontheﬂy as the reachability computation evolves, rather than being precomputed for the whole state space in advance. However, even ontheﬂy generation cannot cope with the fact that in high dimension, a tube of reachable states will typically leave a box via exponentially many facets. This situation is illustrated in Figure 3(a). Since each of these parts of the reachable set goes to a diﬀerent box, they have to be handled separately (Figure 3(b)) even though they continue to evolve close to each other.2 Merging these sets when they converge to the same box is a tedious process and a source of further approximation errors. This problem is particularly severe because making the boxes smaller is the recommended recipe for improving accuracy.
2
A similar phenomenon has been encountered in the analysis of timed automata [9].
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Dynamic Hybridization
In this section we describe our novel nonlinear reachability algorithm which, unlike the scheme of [6], is not based on a ﬁxed partitioning of the state space but rather generates overlapping linearization domains around the reachable states. An important ingredient of any hybridization methodology is the linearization procedure that we ﬁrst deﬁne formally. Definition 4 (Linearization in a Domain). A linearization operator is a function L which, for a given nonlinear function f and a convex set B (linearization domain), produces a matrix A and a convex polytope V such that for every x ∈ B, f (x) ∈ Ax ⊕ V . We use the notation L(f, B) = (A, V ). In our current implementation the linearization domains are boxes, but other forms are possible. In addition to the linearization operator L and the linear successor operator σ we assume a procedure β which takes as input a set P and produces a linearization domain B = β(P ) which contains P . The form of B, the relation between its size and the size of P as well as the position of P inside B are implementation details that may vary according to the system in question. We ﬁrst present in general terms the algorithm for approximates the reachable states, prove its correctness and then discuss our implementation of L and β. Algorithm 2 (Dynamic Hybridization) Input: A nonlinear dynamical system S = (X, f ) and an initial set P Output: A sequence of sets P0 , P1 , . . . Pk whose union includes R[0,h] (P ) B := β(P ) (A, V ) := L(f, B) ˜ [0,r] (P ) P0 := R i := 0 repeat Pi+1 := σ(Pi , A, V, r) if Pi+1 ⊆ B i := i + 1 else B := β(Pi ) (A, V ) := L(f, B) until i = k The algorithm performs linear reachability in a linearization domain B as long as the computed sets remain inside B. Once a newlycomputed set Pi+1 is not fully contained in B we backtrack to Pi and construct a new domain B around Pi along with its corresponding linearization which is used for subsequent computations starting from Pi , as illustrated in Figure 4. The advantage of this approach is obvious: the linearization mesh is constructed along the reachable set and thus we avoid artiﬁcial splitting of sets due to the structure of the mesh. Needless to say, the intersection operation is altogether avoided.
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(a)
(b)
Fig. 3. (a) the reachable set leaves a box through several boundaries; (b) the computation is continued separately for each intersection although the computed sets remain close to each other and even go later to the same box
Pi
Pi
P0
P0
B B
B (a)
(b)
Fig. 4. Dynamic hybridization: (a) Computing in some box until intersection with the boundary; (b) Backtracking one step and computing in a new box
Theorem 1 (Correctness of Algorithm 2). Let P0 , P1 , . . . be a sequence of sets produced by Algorithm 2. Then for every k ≤ k , we have
R[kr,k r] (P ) ⊆
k
Pi .
i=k
Proof. The proof is by induction on the number of switchings between linearization domains that the algorithm makes. The base case where no switching occurs follows from the correctness of the linear reachability algorithm and the fact that the linearized system over approximates f . For the inductive case, assume the claim holds for s switchings and consider a run of the algorithm with s + 1 switchings, the last of which occurring after Pj , k ≤ j < k . By the inductive hypothesis R[jr,jr] (P ) ⊆ Pj and since Pj serves as the initial set for subsequent iterations inside a single linearization domain, the base case applies and Pj+1 , . . . , Pk includes R[(j+1)r,k r] (P ) which, together with Pk , . . . Pj , include the states reachable within [kr, k r]. Algorithm 2 is implemented in C and uses the polytopebased algorithms of d/dt [13]. Below we explain the novel technical aspects, namely the dynamic construction of the linearization domain and its respective linearization. The diﬀerence between the function f and its linear approximation A relative to a domain B is ΔB (f, A) = {f (x) − Ax : x ∈ B}. To guarantee conservative approximation it is suﬃcient to ﬁnd some V such that ΔB (f, A) ⊆ V and this can be done easily for any choice of a domain B and a linearization A. However to
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obtain highquality approximations, we need to choose B and A that minimize, roughly speaking, the diameter of ΔB (f, A) which represents the error incurred by the linear over approximation. Clearly the smaller is B, the smaller is the error but then the linearization procedure has to be invoked more frequently. The problem of ﬁnding good B and A can be formulated, in principle, as some sort of a constrained optimization problem but this computation can be very costly and we use instead the following easytocompute heuristic which turns out to work in practice despite being non optimal. The ﬁrst simpliﬁcation that we do with respect to an optimized solution is to decouple the choice of the new domain B = β(P ) from the computation of the linearization (A, V ) = L(B, f ). The operator β(P ) which produces a box containing P is realized as follows. Based on f and X we ﬁx a standard rectangular frame B of size d1 ×, · · · , ×dn . Given a polytope P we deﬁne its centroid c(P ) to be the average of its vertices and let β(P ) be a copy of B whose center coincides with c(P ). The only problematic situation occurs when during reachability computation P gets too large and cannot ﬁt (either immediately or after few steps) within the frame B. To prevent Algorithm 2 from getting stuck in the else branch, we split P into two or more sets which are then treated separately. In principle, this splitting may lead to state explosion but, in this case, the explosion is due to intrinsic properties of the set of reachable states and not due to an arbitrary choice of the coordinate system underlying the mesh. This phenomenon will not occur too often while analyzing stable systems having a contracting dynamics. To handle the splitting we ﬁrst compute a tight bounding box B(P ) around P . This computation is performed by projecting the vertices on each of the dimensions and taking the minimum and maximum. Let us denote by e1 ×, · · · , ×en the size of the obtained bounding box. If for every i, m · ei < di , where m > 1 is a ﬁxed constant, then P is suﬃciently small and no splitting takes place. Otherwise we take the direction i which maximizes the ratio ei /di and split P into two parts along this direction by intersecting it with complementary halfspaces orthogonal to direction i (see Figure 5). We repeat the process until the obtained sets are suﬃciently small. We thus end up with one or more polytopes around each of which we put a properlycentered copy of B. Once the linearization domain B is ﬁxed we compute A and V as follows. Let f = (f1 , . . . , fn ), and let y = c(B) be the center of B. The matrix A is obtained by the evaluating (numerically) the Jacobian matrix of f at y, that is,
P1 P
P2 B(P ) B2 B1
Fig. 5. A set P and its bounding box B(P ). The set is too large and is split in the vertical dimension into P1 and P2 , around which the respective linearization domains B1 and B2 are constructed.
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∂fi A = ∂f ∂x (y) where Aij = ∂xj . Then a box V = V1 × V2 × . . . × Vn , guaranteed to contain ΔB (f, A), is computed as follows. For each dimension i we let Vi be the interval [li , ui ] where li = min{πi (ΔB (f, A))} and ui = max{πi (ΔB (f, A))} with πi denoting projection on i. These intervals are over approximated based on the Taylor expansion of f (x) − Ax.
4
Experimental Results
To test the feasibility of our algorithm we applied it to two nonlinear systems whose parameters and qualitative behaviors are documented in the literature. The Lac Operon is a biochemical feedback mechanism through which the bacterium E. Coli adapts to the lack of Glucose in its environment by switching to a Lactose diet. We use the model appearing in [31] where the behavior of the system is described by the following system of diﬀerential equations: R˙ a = τ − μ ∗ Ra − k2 Ra Of + k−2 (χ − Of ) − k3 Ra Ii2 + k8 Ri G2 O˙ f = −k2 ra Of + k−2 (χ − Of ) E˙ = νk4 Of − k7 E M˙ = νk4 Of − k6 M I˙i = −2k3 Ra Ii2 + 2k−3 F1 + k5 Ir M − k−5 Ii M − k9 Ii E G˙ = −2k8 Ri G2 + 2k−8 Ra + k9 Ii E
The diﬀerential variables denote the concentrations of diﬀerent reactants, such as Ra (active repressor) Of (free operator), E (enzyme), M (mRNA), Ii (internal inducer), and G (glucose). We studied the behavior of this 6dimensional system around a quasisteady state for the ﬁrst 4 variables and the obtained results are consistent with the simulation results obtained on a simpliﬁed 2dimensional model shown in [31], page 285. As a set of initial states we take a small box where
(a)
(b)
Fig. 6. Lac operon: (a) a stable focus, k−1 = 2.0; (b) a limit cycle, k−1 = 0.008
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Fig. 7. Results obtained for the aging model
Ii ∈ [1.9, 2.0] and G ∈ [25.9, 26]. When k−1 = 2.0 the system exhibits a stable focus and when k−1 = 0.008 the system exhibits a limit cycle (see Figure 6). Computation times are 3 and 5 minutes, respectively. We conclude with a model of an aging process, based on the mitochondrial theory of aging. The highlynonlinear diﬀerential equations, which include ratios between variables, can be found in [31], page 252. The model admits 9 variables and we show in Figure 7 the reachable set after 300 iterations projected on 3 variables, namely, the concentration of antioxidants (AOx), of radicals (RAdM) which suﬀer damages, and of ATP (adenosine triphosphate). After 1000 iterations, we observe the convergence towards a steady state. The computation time for 1000 iterations is 23.3 minutes.
5
Discussion
We made progress toward a very ambitious goal: automatic reachability analysis of nonlinear systems as a methodology for investigating underspeciﬁed biological models. Let us mention other attempts to solve this problem starting with methods that share with hybridization the idea of approximating the original systems by partitioning the continuous state space and producing a hybrid automaton with a simpler dynamics in each state. In the extreme case where no continuous dynamics is left the ﬁnite automaton is the sole responsible for approximating the dynamics. This approach is common in AI and qualitative physics and has been used extensively in Biology [21, 30, 19]. The technique of predicate abstraction applied to hybrid systems [3] is another elaboration of this idea where partition boundaries are based on predicates appearing in speciﬁcations. A more reﬁned approach, incorporated into the tools HyTech [20] and PHAVer [17] over approximates the nonlinear system by hybrid automata where in each state the dynamics is deﬁned by a constant diﬀerential inclusion of the form Ax˙ ≤ c. Since in each state the derivative does not depend on the real variables, it is easy to compute the reachable states exactly using linear algebra, however the over approximation with respect to the original system is large (zeroorder
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compared to ﬁrstorder approximation in the hybridization of [6]). The translation of continuous systems into timed automata [36] is another example. Other, more direct, approaches perform reachability on the original nonlinear systems without relying on convexity properties. For example, the face lifting technique [25, 15, 26], which is based on computing the maximal projections of f on all the normals of the facets of a polyhedron, may lead to large overapproximation errors. Other approaches use more complex classes of sets which are not necessarily convex. In [37] the evolution of the reachable states is transformed into a partial diﬀerential equation (PDE) where the boundary of the set is represented as the set of zeros of a function deﬁned over the state space. The work of [14] uses Bezier simplices to represent reachable states for systems deﬁned by polynomial diﬀerential equations. Finally in [18,1] dynamic linearization and computation of error bounds is performed at every reachability step. None of these methods, to the best of our knowledge, can cope with systems of the size and complexity of the examples presented in this paper. Let us also mention the whole domain of interval analysis [40], a branch of numerical analysis motivated by producing rigorous numerical answers to diverse mathematical questions despite roundoﬀ errors. As its name suggests, for the computation of a scalar function, the result is typically an interval guaranteed to contain the correct answer. The generalization to many dimensions leads naturally to bounding boxes. Although the motivation is diﬀerent from ours as the uncertainty is due to the computation itself rather than the imperfection of the model, there are similarities between some of the techniques and we foresee more future cross fertilization between the domains. Parameter uncertainty in biological models is a wellknown problem that has been subject to extensive work using various techniques. We mention two recent attacks on the problem of parameter synthesis, namely, ﬁnding or approximating the range of model parameters for which some qualitative behavior is exhibited. The work of [8] takes a hybrid model (piecewise multiaﬃne dynamics) with parameter uncertainty and abstracts it into a ﬁnite automaton. When the property in question is violated by the automaton, the domain of parameter values is reﬁned, a new abstraction is created and so on. A more direct and eﬃcient way to explore the space of parameter values is described in [16] based on adaptive sampling of the parameter space and using ordinary numerical simulation. This technique uses numerical sensitivity information to guide the reﬁnement of the parameter space. To go further we need to combine dynamic hybridization with the algorithm of [24,27] which can treat linear systems an order of magnitude larger than those treated in the present paper. To this end we need to develop a good splitting procedure for the sets computed by that algorithm. As the reader might have noticed, we have focused here on systems where the uncertainty is restricted to the initial set and we need to extend our linearization operator to nonlinear functions with input, something that can be done using similar principles. To conclude, we have demonstrated the feasibility of our approach by computing reachable states for nonlinear systems of unpreceded size and complexity.
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We intend to pursue this direction further and make reachability computation a useful tool for analyzing complex biological systems. A parallel eﬀort should be invested in making modelers of biological systems aware of the potential of this analysis technology.
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On Coupling Models Using ModelChecking: Eﬀects of Irinotecan Injections on the Mammalian Cell Cycle Elisabetta De Maria, Fran¸cois Fages, and Sylvain Soliman Projectteam Contraintes, INRIA ParisRocquencourt, France
Abstract. In systems biology, the number of models of cellular processes increases rapidly, but reusing models in diﬀerent contexts or for diﬀerent questions remains a challenging issue. In this paper, we show how the validation of a coupled model and the optimization of its parameters with respect to biological properties formalized in temporal logics, can be done automatically by modelchecking. More speciﬁcally, we illustrate this approach with the coupling of existing models of the mammalian cell cycle, the p53based DNAdamage repair network, and irinotecan metabolism, with respect to the biological properties of this anticancer drug.
1
Introduction
In systems biology, the number of models of cellular processes increases rapidly. To date, most of the eﬀort has been devoted to building models and making them freely available, through the design of standard exchange formats, such as for instance the Systems Markup Language SBML [26], the making of model repositories, such as for instance http://biomodels.net/, the making of biological ontologies to establish the links between molecular synonyms, species, units, etc., and the development of modeling tools, such as Cell Designer, BIOCHAM [7], BioNetGen [4], Pathway Logic [15], Bioambients [35], etc. Despite these eﬀorts however, reusing models in diﬀerent contexts or for diﬀerent questions remains a challenging issue. In practice, most of the models are developed, reﬁned, simpliﬁed or coupled with respect to other models by hand with no direct support from the tools to reuse models in a systematic way using a speciﬁcation of the expected behavior. In this paper, we show how the validation of a coupled model and the optimization of its parameters with respect to biological properties formalized in temporal logics, can be done automatically by modelchecking. More speciﬁcally, we illustrate this approach with the coupling of existing models of the mammalian cell cycle, the p53based DNAdamage repair network, and irinotecan metabolism, with respect to the biological properties of the latter one. Irinotecan is an anticarcinogenic inhibitor of topoisomerase1 which started to be used in clinical treatments approximately twenty years ago. It shows signiﬁcant eﬃcacy against a variety of solid tumors, including lung, colorectal, P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 142–157, 2009. c SpringerVerlag Berlin Heidelberg 2009
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and cervical cancers. Scientists are currently trying to optimize the irinotecan therapy in order to understand how to limit its toxicity and to increase its eﬃcacy. In this context, it is crucial to comprehend how the presence of this medicament inﬂuences cellular proliferation. In this work we present a modelchecking approach to the problem. There are in the literature many models of the mammalian cellcycle, a few ones of the cell’s DNAdamage repair pathways, and recently some preliminary models of irinotecan intracellular pharmacodynamics. However these modules need to be assembled in a coherent way to provide meaningful answers. After reformulating the existing models in the rulebased language of the biochemical abstract machine Biocham [19,7], we assemble them into a coupled model under the constraint of satisfying some relevant biological properties, formalized in temporal logic and automatically checked by modelchecking. Modelchecking is the process of algorithmically verifying whether a given structure is a model for a given temporal logic formula [13]. In literature, there are various applications of modelchecking techniques to biology. In [8,15], temporal logic was introduced as a query language for biochemical networks and for validating boolean models of biological processes by modelchecking techniques. Some experimental results were obtained on Kohn’s map of the mammalian cell cycle control (800 reaction rules, 500 variables) using the symbolic modelchecker NuSMV, and on a small ordinary diﬀerential equation (ODE) model using the constraintbased model checker DMC. This approach to verifying biological processes has pushed the development of modelchecking techniques for quantitative properties, and continuous, stochastic or hybrid models. For (nonlinear) ODE models, numerical integration techniques provide numerical traces on which linear time temporal logic with numerical constraints can be evaluated by modelchecking. Simpathica [2] and the biochemical abstract machine Biocham [6] are two examples of computational tools integrating such modelcheckers for quantitative models. This approach has been generalized to temporal logic constraint solving in [18], allowing for eﬃcient kinetic parameter optimization [37] and robustness analyses [36] w.r.t. quantitative temporal logic properties. In [25], Heath et al. apply the probabilistic modelchecker PRISM to the study of a complex biological system, namely, the Fibroblast Growth Factor (FGF) signalling pathway. In [12], Clarke et al. apply statistical modelchecking on a stochastic model of a Tcell receptor. In [3] Batt et al. develop a modeling framework based on diﬀerential equations to analyze genetic regulatory networks with parameter uncertainty. The values of uncertain parameters are given in terms of intervals and dynamical properties of the networks are expressed in temporal logic. Modelchecking techniques are then exploited to prove that, for every possible parameter value, the modeled systems satisfy the expected properties and to ﬁnd valid subsets of a given set of parameter values (such an approach is exploited in RoVerGeNe, a tool for robust veriﬁcation of gene networks). In [33], Piazza et al. propose semialgebraic hybrid systems as a natural framework for modeling biochemical networks, taking
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advantage of the decidability of the modelchecking problem for TCTL (Timed Computation Tree Logic) over this large class of systems. In this paper, we focus on the use of modelchecking for integrating biological models. To compose the selected models, we assume a ﬁnite set of hypotheses concerning the structure of the “linking reaction rules”, and we search for kinetic parameter values that will make the composite model interacts in a proper way. For this, the biological properties of the coupled model are formalized in temporal logic with numerical constraints, and modelchecking with parameter optimization techniques are used to ﬁnd parameter values for the new kinetic rules so that the expected properties are satisﬁed by the resulting model. The paper is organized as follows. Section 2 provides the needed biological background on the mammalian cell cycle, on the tumorsuppressor protein p53, and on irinotecan. Section 3 describes Biocham models and explains how such models can be queried in temporal logic. Section 4 presents our coupled model on the eﬀects of irinotecan and Section 5 describes the biological properties of the model that were automatically checked. Finally, in Section 6 we outline some ongoing developments of this work. All the models used are available in BIOCHAM format at http://contraintes.inria.fr/supplementary material/CMSB09/.
2 2.1
Mammalian Cell Cycle, DNADamage Repair and Irinotecan Mammalian Cell Division Cycle
Cells reproduce by duplicating their contents and then dividing in two. To produce a pair of genetically identical daughter cells, the DNA must be faithfully replicated, and the replicated chromosomes must be segregated into two separate cells. The duration of the cell cycle varies greatly from one cell type to another; in a mammalian cell it lasts about 24 hours. The cycle is traditionally divided into the following four distinct phases [1]: the G1phase, that is the temporal gap between the completion of mitosis and the beginning of DNA synthesis, the Sphase (synthesis), that is the period of DNA replication, the G2phase, that is the temporal gap between the end of DNA synthesis and the beginning of mitosis, and the Mphase (mitosis), when replicated DNA molecules are ﬁnally separated in two daughter cells. The cell cycle is regulated by diﬀerent checkpoints, that are moments when the cell progression is stopped to verify the state of the cell and, if needed, to repair it before damaged DNA is transmitted to progeny cells. DNA damaging agents trigger checkpoints that produce arrest in G1 and G2 stages of the cell cycle. Cells can also arrest in S, which amounts to a prolonged S phase with slowed DNA synthesis. Arrest in G1 allows repair before DNA replication, whereas arrest in G2 allows repair before chromosome separation in mitosis. The proper alternation between synthesis and mitosis is coordinated by a complicated network that regulates the activity of a family of key proteins. These proteins are composed of two subunits: a catalytic subunit, the cyclindependent kinase, cdk for short, and a regulatory subunit, a cyclin. A cdk has to associate
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with a cyclin partner to form a dimer and has to be appropriately phosphorylated in order to be active. The progression through cell cycle is orchestrated by the rise and fall of the Cdk/cyclin dimers. In this work we refer to the model of mammalian cell division proposed by Nov´ak and Tyson in [30] where the authors present both a set of Ordinary Differential Equations (ODE) and a process diagram to represent the molecular network regulating the mammalian cell cycle. The model comprises 18 diﬀerential equations and 4 steadystate relations. 2.2
Protein p53
This subsection is devoted to the description of protein p53, a tumor suppressor protein which is activated in reply to DNA damage. In normal conditions, the concentration of p53 in the nucleus of a cell is feeble: its level is controlled by another protein, Mdm2. These two proteins present a loop of negative regulation. In fact, p53 activates the transcription of Mdm2 while the latter accelerates the degradation of the former. DNA damage increases the degradation rate of Mdm2 so that the control of this protein on p53 becomes weaker and p53 can exercise its functions. This protein is responsible for the activation of many mechanisms: in an indirect way, it stops the DNA synthesis process, it activates the production of proteins charged with DNA reparation, and it can lead to apoptosis, that is, cell death. When DNA is damaged, Mdm2 loosens its inﬂuence on p53 and it is possible to observe some oscillations of p53 and Mdm2 concentrations. The answer to a stronger damage is a bigger number of oscillations. Oscillations have a very regular period. In literature, several models have been proposed to model the oscillatory behaviour of proteins p53 and Mdm2. The most interesting models are undoubtedly the ones proposed by Chickermane et al. [9], by Ciliberto et al. [10], and by GevaZatorsky et al. [20]. In this work we build upon the one described in [10], that consists of 6 diﬀerential eqiations. 2.3
Irinotecan
Camptothecins are substances that can be extracted from the Chinese tree “Camptotheca acuminata Decne” and are mainly used for the treatment of digestive cancers. Their anticancerogenic properties have been discovered at the end of the Fifties but the ﬁrst clinical tests have been interrupted owing to heavy eﬀects due to the toxicity of the substances. In the Eighties researchers discovered that camptothecins are inhibitors of topoisomerase1(Top1 for short), essential enzyme for DNA synthesis. Afterwards, they started to focus on some semisynthetic derivative of watersoluble camptothecins, such as irinotecan and topotecan. Irinotecan is promedicine and must be transformed in its active metabolite, SN38, to be eﬀectively cytotoxic. In fact the anticancerogenic activity of irinotecan (CPT11) is approximately 100 times less eﬀective than the one of SN38. The activation is due to carboxylesterase, an enzyme mainly located in the liver, in the intestine, and in the tumoral tissues. SN38 is then detoxiﬁed
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through glucoronoconjugation: this realizes uridine diphosphate glucoronosyl transferase 1A1. Mechanisms through which irinotecan damages the cell are very complex and have not completely been explained yet. It is sure that DNA lesions appear after the inhibition of Top1 by SN38. Top1 is a protein which is present in all living organisms and which checks DNA replication and transcription. It intervenes to modify the DNA winding degree, acting on one strand. More precisely, Top1 links itself to extremity 3’ of DNA forming a transitory cleavage complex and cuts a DNA strand, that in such a way is able to unroll. Then such a complex dissociates and a new ligature comes up. In normal conditions, the connection process is favored with respect to the cleavage one. The target of irinotecan, and above all of its active metabolite SN38, is the complex Top1DNA. SN38 links to the complex through a covalent bond, preventing in such a way from the ligature of the DNA strand. As clearly written in the title of [34], SN38 acts like a “foot in the door”: it keeps opened the DNA strand to which Top1 is linked as to prevent a door from closing. These complexes are still reversible and do not cause DNA lesions. However, they favor them: some lesions can rise as a consequence of the possible collisions with the transcription complexes or with the replication fork. This induces the arrest of the cell cycle. In this case we speak of irreversible complexes. Lesions due to the inhibition of Top1 are therefore consecutive to the stages of the cell metabolism. It means that irinotecan injections must be repeated and abundant in order to be eﬀective. Besides irinotecan is more eﬀective during the DNA replication phase [31,38]. Furthermore, the inhibition of the DNA synthesis takes rapidly place (in a few minutes) and lasts several hours. Defence answers of cells subjected to irinotecan injections are multiple and vary according to the drug dose. The administration of a very light dose suﬃces to slow down the S phase of the cell cycle and to delay the G2M transition. If the dose is more substantial, the lag time in the S phase is much more signiﬁcant and the cell cycle arrest in the G2M transition can last more than sixty hours or even be permanent. In this latter case, some genes responsible for the cell cycle arrest (as an example, p21) and involved in the aptototic pathway are overexpressed. These genes are activated by p53, and this suggests the intervention of the protein in reply to a DNA damage due to the dissociation of Top1 from DNA [38]. In this work we refer to a pharmacokinetics/pharmacodinamics (PK/PD) model of irinotecan developed by Dimitrio [14], that takes aim at representing the action of the drug on the body (pharmacodinamic) and the action of the body on the drug (pharmacokinetic), and thus the drug metabolism and its transformations. Such a model is made up of 8 diﬀerential equations.
3
The BIOCHAM Abstract Machine
In the last years, one of the main challenges of computational system biology became the creation of powerful simulation, analysis, and reasoning tools for
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biologists to decipher existing data, devise new experiments, and thus understand functional properties of genomes, proteomes, cells, organs, and organisms. Many of the goals of this emerging discipline have been investigated in the logical setting of temporal logics and have been partly achieved in this apprroacch in the Biochemical Abstract Machine (BIOCHAM ) [19,7], a formal modeling environment for network biology developed at INRIARocquencourt since 2002. In this section we brieﬂy recall BIOCHAM syntax, its continuous diﬀerential semantics and show how a temporal logic with numerical constraints is used to formalize the biological properties of the models and automatically check their satisfaction. 3.1
Syntax and Diﬀerential Semantics of BIOCHAM Reaction Models
Following SBML and BIOCHAM conventions, a model of a biochemical system is formally a set of reaction rules of the form e for S => S , where S is a solution, that is, a set of molecules given with their stoichiometric coeﬃcient, S is the transformed solution, and e is a kinetic expression involving the concentrations of molecules. The reaction rules represent biomolecular interactions between chemical or biochemical compounds, ranging from small molecules to proteins and genes. Reaction rules transform one formal solution into another one. The following abbreviations are used: A =[C]=> B for the catalyzed reaction A+C => C+B, and A B for the reversible reaction equivalent to the two symmetrical reactions A => B and B => A. The constant represents the empty solution. It is used for instance in protein degradation rules, such us A => , and in synthesis rules, such us =[G]=> A for the synthesis of A by (activated gene) catalyst G. The other main rule schemas are (de)complexation rules, such us A + B => AB for the complexation of A and B, (de)phosphorylation rules, such us A =[B]=> A~{p} for the phosphorylation of A catalyzed by kinase B, and transport rules, such us A::nucleus => A::cytoplasm for the transport of A from the nucleus to the cytoplasm. Reactions can be given kinetic expressions. For instance, k*[A]*[B] for ~ speciﬁes a mass action law kinetics with parameter k for the reacA=[B]=>A{p} tion. Classical kinetics expressions are the following ones: n – the mass action law kinetics k ∗ i=1 xlii , which refers to a reaction with n reactants xi , where li is the stoichiometric coeﬃcient of xi as a reactant; – the MichaelisMenten kinetics Vm ∗ xs /(Km + xs ), which states for an enzymatic reaction of the form xs = [xe ] => xp , where1 Vm = k ∗ (xe + xe ∗ xs /Km ); n – the Hill’s kinetics Vm ∗ xns /(Km + xns ), of which the MichaelisMenten kinetics is a special case for n = 1. 1
xe ∗xs /Km is the concentration of the enzymesubstrate complex, supposed constant in the Michaelian approximation, and xe + xe ∗ xs /Km is thus the total amount of enzyme.
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Kinetic expressions can be written either explicitly, allowing any kinetics, or using shortcuts such us MA(k) for a Mass Action law with parameter k, or MM(Vm,Km) for a Michaelian kinetics. A set of reaction rules {ei for Si => Si }i=1,...,n over molecular concentration variables {x1 , . . . , xm } can be interpreted under diﬀerent semantics. In this paper we refer to the traditional diﬀerential semantics, that interprets the rules by the following system of Ordinary Diﬀerential Equations (ODE): dxk /dt =
n i=1
ri (xk ) ∗ ei −
n
lj (xk ) ∗ ej ,
j=1
where ri (xk ) (resp., li ) is the stoichiometric coeﬃcient of xk in the right (resp., left) member of rule i. Given a set of reaction rules, BIOCHAM allows to obtain a simulation of the model by solving the set of corresponding diﬀerential equations. 3.2
Querying BIOCHAM Models in Temporal Logic
Temporal logics and modelchecking algorithms have been proved to be useful to respectively express biological properties of complex biochemical systems and automatically verify their satisfaction [17]. Having a formal language not only for describing models, (i.e., transition systems based on process calculi [35,32], rules [15], Petri nets [22], ODEs [39], etc.), but also for formalizing the biological properties of the system known from biological experiments under various conditions, opens a whole avenue of research for designing automated reasoning tools inspired from circuit and program veriﬁcation to help the modeler. In this paper, we use a version of linear time logic LTL with numerical constraints, named ConstraintLTL [6]. ConstraintLTL formulae are formed over ﬁrstorder atomic formulae with equality, inequality and arithmetic operators ranging over real values of concentrations and of their derivatives, using the logical connectives and the usual temporal operators of LTL: in particular the “always in the future” operator G, and the “sometimes in the future” operator F), the next time operator X, and the binary operator until U. For instance, F ([A] > 10) expresses that the concentration of A eventually gets above the threshold value 10 and G([A] + [B] < [C]) states that the concentration of C is always greater than the sum of the concentrations of A and B. Oscillation properties, abbreviated as oscil(M, K), are deﬁned as a change of sign of the derivative of M at least K times: F ((d[M ]/dt > 0) & F ((d[M ]/dt < 0) & F ((d[M ]/dt > 0) . . . ))). The abbreviated formula oscil(M, K, V ) adds the constraint that the maximum concentration of M must be above the threshold V in at least K oscillations. LTL formulae are interpreted in linear Kripke structures which represent either an experimental data time series or a simulation trace, both completed with loops on terminal states. Given the system of ordinary diﬀerential equations (ODE) corresponding to a reaction model, under the hypothesis that the initial state is completely deﬁned, a discrete simulation trace is easily obtained by means of numerical integration methods (such as RungeKutta or Rosenbrock
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method for stiﬀ systems). Since constraints refer not only to concentrations, but also to their derivatives, traces of the form (< t0 , x0 , dx0 /dt, d2 x0 /dt2 >, < t1 , x1 , dx1 /dt, d2 x1 /dt2 >, . . . ) are considered, where at each time point ti , the trace associates the concentration values xi to the variables, and the values of their ﬁrst and second derivatives dxi /dt and d2 xi /dt2 . It is worth noting that in adaptive step size integration methods of ODE systems, the step size ti+1  ti is not constant and is determined through an estimation of the error made by the discretization. Observe that the notion of next state refers to the state of the following time point in a discretized trace, and thus does not necessarily imply a real time neighborhood. The rationale is that the numerical trace contains enough relevant points, and in particular those where the derivatives change abruptly, to correctly evaluate temporal logic formulae. An innovative feature of BIOCHAM is that it places at the user’s disposal a procedure learn parameters for ﬁnding parameter values such that a given LTL formula is satisﬁed [37]. This search procedure actually replicates and automates part of what the modeler currently does by hand: trying diﬀerent parameter values, between bounds that are thought reasonable, or computed by other methods such as bifurcation diagrams, in order to obtain behaviors in accordance with the experimental knowledge.
4
4.1
Coupling the Models of the Mammalian Cell Cycle, p53Based DNADamage Repair System, and Irinotecan Pharmacodynamics Three Diﬀerent Modules
The ﬁrst step of our approach to the investigation of the inﬂuence of irinotecan on the mammalian cell cycle consists in the encoding of the selected models of irinotecan [14], p53/Mdm2 [10], and mammalian cell cycle [30] in the BIOCHAM rulebased language. SBML versions of the irinotecan and p53/Mdm2 modules being available, they were imported in BIOCHAM. The renaming of the variable representing DNAdamage was the only modiﬁcation necessary in this precise case. More generally it would be necessary to rely on existing databases and ontologies to match corresponding entities in diﬀerent models. For the other model, we looked in parallel at the corresponding set of ordinary diﬀerential equations and at the available diagrammatic notation to obtain a set of BIOCHAM reaction rules. Since ODEs can be automatically extracted back from the reactions, and displayed with the BIOCHAM command show kinetics, it was possible to check that the obtained model was indeed coherent with the original one. 4.2
A Diagrammatic Coupled Model
In order to assemble the submodels to get the coupled model, we reviewed the literature about known links between the diﬀerent building blocks.
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The expected behaviour of the cell is graphically depicted in Figure 1 and can be described as follows. Injections of irinotecan (CPT11) induce DNA damage. In reply to this, the cell reacts by activating protein p53, which blocks the cell cycle at a checkpoint. This arrest aims at repairing critical damage before DNA replications occurs, thereby avoiding the propagation of genetic lesions to progeny cells. Thus, while the cell cycle is arrested, the protein p53 will activate the DNAdamage repair mechanisms. If it is possible for the cell to recover, the cell cycle will be restarted; otherwise, if the damage is too extensive, the cell will undergo apoptosis. The strategy we chose to draw the three models together is illustrated in Figure 2. As remarked in Section 2, in literature we found evidence of the fact that, if irinotecan is injected in a cell during the S phase of the cell cycle, then more DNA damage will be caused with respect to the other phases of the cell cycle [31,38]. Keeping this fact in mind, and considering that the dimer characterizing S phase is CycA/Cdk2 (CycA for short), we inserted in cell cycle model a rule stating that a high concentration of CycA determines a high concentration of Top1, an enzyme that, as previously explained, contributes to cause DNA singlestrand breaks in presence of irinotecan. In this way we linked the cell cycle model to the irinotecan one. The link between the irinotecan model and the p53/Mdm2 one is given by DNA damage. In fact, irinotecan injections cause DNA damage, which in turn triggers the activity of protein p53. In order to link the p53/Mdm2 and cell
CPT11
DNA damage
Block cell cycle at checkpoints
p53
Apoptosis
recover Fig. 1. Expected behaviour of the coupled model CycA
Cell cycle
Top1 Irinotecan
CycA CycE DNA damage
p21 p53 p53/Mdm2
Fig. 2. Our coupled model
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cycle models, we inserted in the p53/Mdm2 model a rule which ﬁxes that p53 activates p21, and two further rules imposing that p21 inhibits CycA and CycE, respectively. It is worth noting that we also investigated the possibility to abstract the previous expanded rules by letting p53 directly inhibit CycA and CycE. In the following, we will refer to this last version of the link as to the contracted one. Finally, in order to deﬁne precisely the kinetics underlying the links and to validate the coupled model, some biological knowledge was formalized as temporal logic speciﬁcations allowing us to use modelchecking and parameter optimization techniques.
5
Specifying and Validating the Links through ModelChecking
In this section, we will show how the integration of modelchecking and parameter learning techniques allowed us to both specify and validate our linking rules. In both cases, we took advantage of constraintLTL formulae to query numerical simulations in a much more ﬂexible way than by doing curve ﬁtting. Our Kripke structures are constituted of simulation traces over a time window of 100 hours, obtained by numerical integration (using Rosenbrock’s implicit method for stiﬀ systems), extended with the ﬁrst and second derivatives of the system’s variables. As a matter of fact, both at speciﬁcation and validation time, ﬁrst we considered two models at a time, and ﬁnally we dealt with the complete model made up by the three ones. In the following, we provide examples of biological properties that helped us in deﬁning the kinetics underlying the links: for each property, ﬁrst of all we express it through the natural language, then we formalize it in constraintLTL, and ﬁnally we present the results of the corresponding modelchecking query. We encoded the link between the cell cycle and irinotecan models (see Figure 2) by means of the following kinetics rule: MA(top1bis) for =[CycA]=>TOP1. Then, we used the procedure learn parameters to ﬁnd out the minimum value for top1bis such that property F1, that correlates the concentration values of CycA and Top1, holds. F1: Whenever CycA gets over 1, there exists a future state where Top1 is greater than 1. LTL: G(([CycA] > 1) → F ([T OP 1] > 1)). Results: The minimum value for top1bis such that F1 is satisﬁed turned out to be 0.45. The following Biocham rules encode the link between the p53/Mdm2 and cell cycle models (see Figure 2): MA(k5321) for =[p53]=>p21. MA(kA21) for CycA=[p21]=> .
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MA(kA21) for CycE=[p21]=> . As for the contracted version, the encoding is the following one: MA(kA53) for CycA=[p53]=> . MA(kA53) for CycE=[p53]=> . Again, we took advantage of the procedure learn parameters to ﬁnd suitable parameter values for k5321 and kA21 (kA53 in the second case) so that property F2, that expresses the CycA oscillating behaviour exhibited in the cell cycle model, is conserved in the coupled model in case there are no irinotecan injection. F2: Within a time interval of 100 time units, CycA is greater than 2 in at least 8 oscillations. LTL: oscil([CycA], 8, 2). Results: We found out that suitable parameter values are k5321 = 0.18, kA21 = 0.08, and kA53 = 0.25. Property F2 also turned out to be true when there are injections but the p53/Mdm2 model is not taken into account. In fact, as expected, even if DNA damage occurs, when protein p53 does not act, the cell cycle is not aﬀected, and thus CycA exhibits a regular oscillating behaviour. On the other hand, when the p53/Mdm2 model is added, in case of repeated injections (and thus of sustained DNAdamage) the oscillations of CycA should be aﬀected. F3: When there is sustained DNA damage (after an initial period), the amplitude of CycA decreases before 70 time units and then stays low. LTL: F ((T ime < 13) ∧ G([DN Adam] > 0.5)) → F ((T ime < 70) ∧ G([CycA] < 1.2)). Results: With the expanded version of the links the amplitude of oscillations gradually decreases, satisfying the property. With the contracted one oscillations are very irregular, as graphically depicted in Figure 3. The next property regards the DNA repairing power of the cell. F4: After an irinotecan injection is performed, DNA damage is able to go under the threshold of 0.1 before the next injection is done. LTL: G(([CP T 11] > 9.45) ∨ (([CP T 11] ≤ 9.45)U ([DN Adam] < 0.1))). Results: Before testing the property, we decided to parameterize the lapse of time between consecutive irinotecan injections. Then we took advantage of the procedure learn parameters to ﬁnd the minimum k such that, if one injection is performed every k hours, then property F4 is true. We found out that the minimum k multiple of 12 which makes F4 true is 36. Thus, one injection every 36 hours should be performed in order to allow DNA damage to be recovered before the next injection. Then we tried to see what it happens if, at each injection, we double the irinotecan dose. In this case, one injection every 48 hours should be done. The next property requires the oscillating trend of proteins p53 and Mdm2 to stop before a new injection is performed. F5: When an injection is performed, p53 and Mdm2 are in a steady state, that is, their derivatives approach 0. LTL: G(([CP T 11] > 9.45) → ((d[p53] ≤ 0.05) ∧ (d[p53] ≥ −0.05) ∧ (d[M dm2 :: n] ≤ 0.05) ∧ (d[M dm2 :: n] ≥ −0.05))).
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TOP1 p53 Mdm2::n CycA CPT11 DNAdam
2.5
2
1.5
1
0.5
0 0
20
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Fig. 3. Zoomed simulation plot of the resulting model with injections every 36 hours
Results: As for the previous speciﬁcation, we parameterized the lapse of time between consecutive irinotecan injections and we used the procedure learn parameters. The the minimum k multiple of 12 which makes F5 true is 48. The next property deals with DNA damage. F6: If p53 is not functional, DNA damage is an increasing function. LTL: G([p53] ≤ 0.1) → G(d([DN Adam])/dt ≥ 0). Results: As matter of fact, the premise of this property only holds when the p53/Mdm2 model is not taken into account, thus DNA damage is not recovered and continuously increases until apoptosis is reached. Finally, the last two properties concern the oscillating trend of proteins p53 and Mdm2 caused by irinotecan injections. F7: An irinotecan injection causes at least one oscillation of proteins p53 and Mdm2. LTL:G(([CP T 11] > 9.45) → F (oscil([p53], 1) ∧ F (oscil([M dm2], 1)))). F8: p53 oscillations are alternated by Mdm2 ones. LTL: G(oscil([p53], 1) → X((¬oscil([p53], 1))U (oscil([M dm2 :: n], 1)))). Results: Both F7 and F8 are satisﬁed in the coupled model. Such a modelchecking approach turned out to be very eﬃcient: the execution times were lower than one second in almost all cases2 . Furthermore, it proved to be eﬀective, allowing us to express relevant biological properties of the model (and concentration values that make speciﬁcations true) that could not be easily encoded as curve ﬁtting problems for instance. There is a common recognition of 2
All the experiments have been run on a Centrino Duo 2.00 GHz Windows machine.
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the potentialities of modelchecking in bioinformatics and our contribution goes in this direction, showing its utility to compose and validate biological models. In Figure 3 there is a 100 hourssimulation of the complete model, where the contracted version of the link between the p53/Mdm2 and cell cycle models is considered and one injection every 36 hours is performed. The plot puts in evidence how DNA damage increases after every injection. The oscillating trend of proteins p53 and Mdm2 is well highlighted. Furthermore, it is possible to notice the irregular behaviour assumed by CycA after irinotecan injections and the dependence of Top1 from CycA.
6
Conclusion and Perspectives
In this paper we presented a modelchecking approach to the investigation of the inﬂuence of irinotecan on the mammalian cell cycle. We coupled in the rulebased language of BIOCHAM three models of respectively the mammalian cell cycle, p53based DNAdamage repair, and irinotecan intracellular PK/PD, using BIOCHAM’s procedure for optimizing parameters with respect to biological properties formalized in temporal logic. Modelchecking techniques proved to be particularly suitable for studying the irinotecan metabolic pathways. It let us get a better understanding of the drug inﬂuence on the mammalian cell cycle and infer some properties to be exploited in the drug therapy, such us optimal injection times and doses. In order to study how irinotecan interferes with tumor cell proliferation, a further component should be taken into account, that is, the circadian clock, which regulates the synchronous progression of cells through each stage of the cell cycle, determining the daily time windows during which cells can traverse certain phases of the cell cycle [28]. The circadian synchronization of cell cycle progression, which characterizes healthy normal tissues, is often altered in cellular tissues aﬀected by malignant tumors [29]. Roughly speaking, it means that, unlike safe cells, diﬀerent tumor cells at the same time can be in diﬀerent phases of the cell cycle. As a matter of fact, the behaviour of a single cell is not sensibly aﬀected by malignant tumors, but the presence of a disease is generally detected by observing groups of at least one hundred cells. At the level of single cells, a slowing down of biochemical reactions is the only observed phenomenon in presence of tumor. An interesting extension of the work so far presented would consist in adding to our coupled model a new “circadian clock module” to synchronize cells and simulating the resulting model on a group of about one hundred cells. The three already present modules would remain almost unchanged, apart from a speed reduction of the reactions inserted in Figure 2 to link them.
Acknowledgements This work is partly supported by EU FP6 STREP project TEMPO on cancer chronotherapies. We acknowledge fruitful discussions with the partners of this project.
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The κLattice: Decidability Boundaries for Qualitative Analysis in Biological Languages Giorgio Delzanno1 , Cinzia Di Giusto2 , Maurizio Gabbrielli2 , Cosimo Laneve2 , and Gianluigi Zavattaro2 1
Dipartimento di Informatica e Scienze dell’Informazione, Universit` a di Genova, Italia 2 Dipartimento di Scienze dell’Informazione, Universit` a di Bologna, Italia
Abstract. The κcalculus is a formalism for modelling molecular biology where molecules are terms with internal state and sites, bonds are represented by shared names labelling sites, and reactions are represented by rewriting rules. Depending on the shape of the rewriting rules, a lattice of dialects of κ can be obtained. We analyze the expressive power of some of these dialects by focusing on the thin boundary between decidability and undecidability for problems like reachability and coverability.
1
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For this reason, as in other applications of concurrency, an important foundational issue is the study of dialects for which qualitative analysis is computable in an eﬀective way and the isolation of minimal fragments in which it is proved to be impossible. κ is a formalism for modelling molecular biology where molecules are terms with internal state and with sites, bonds are represented by names that label sites, and reactions are represented by rewriting rules. For example, EGFR[tk 0 ](1z ) represents a molecule of species EGFR that is not phosphorilated – the internal state tk is 0 – and that is bond to another molecule – its site 1 is labelled with a name z. The reaction in Fig. 1 deﬁnes the ﬁrst step of the Receptor Tyrosine Kinase (RTK) growth factor EGF (a dimeric form of EGF binds two receptors EGFR, thus phosphorylating the tyrosine kinase site – tk switches from 0 to 1). This reaction is rendered by the following κ rule: EGF (1x + 2y ), EGF (1x + 2z ), EGFR(1y ), EGFR[tk 0 ](1z ) EGF (1x + 2y ), EGF (1x + 2z ), EGFR(1y ), EGFR[tk 1 ](1z )
(1)
Qualitative problems, such as reachability of a given solution, turn out to be undecidable in κ. Therefore one is either compelled to design approximated analyses or to study these properties in dialects of κ. We choose the second direction, thus yielding a number of precise analyses that do not abstract away either from the multiplicity of molecules or from the exact structure of complexes. P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 158–172, 2009. c SpringerVerlag Berlin Heidelberg 2009
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Fig. 1. Representation of the κrule (1)
To this aim, we consider a number of κ dialects that, as we discuss in the following, take inspiration from biological phenomena such as the molecular selfassembly [1] or the DNA branch migration [2]. These dialects are ordered into a lattice by the sublanguage relation – see Figure 2 disregarding the ovals. Let us unravel the lattice with the restrictions imposed on κ to obtain the sublanguages κ−n , κ−d , and κ−d −u . The calculus κ−n follows by removing any form of destruction of molecules (the number of molecules never decrease). This fragment naturally models those systems where molecules always keep their “identity” even when they are part of a complex because, for example, they can subsequently dissociate from the complex. This is the case of polymers, that is chemical structures obtained by joining monomers that react on complementary surfaces. A simple polymerisation – the linear bidirectional one, where the complementary surfaces of monomers are two (that we respectively call l and r in the following) – is modelled by the following κ−n rules: A(r), A(l) A(rx ), A(lx )
(2)
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The reaction (2) deﬁnes polymerization (the creation of a bond between two monomers with free complementary surfaces); (3) deﬁnes depolymerization (the destruction of the bond, but not of the monomers). The additional restriction yielding κ−d is the one that disallows the removal of bonds (depolymerizations are forbidden). This restriction is inspired by molecular selfassembly, which is a process where molecules, initially unbound, adopt a deﬁned arrangement. The DNAorigami method is a popular example of selfassembly that allows to create arbitrary twodimensional shapes, such as Borromean rings [3], using DNA. In κ−d selfassembly is directly enforced because bonds cannot be broken. The last dialect along this axis, called κ−d −u , is obtained by considering molecules without internal states. In several cases such states are not useful. An example is the DNA selfassembly governed by the WatsonCrick complementary base pairing [4]. We also consider two other subcalculi that forbid destructions of molecules and bonds: κ−d−i and κ−d−u−i . These dialects are obtained from κ−d and κ−d −u , respectively, by restricting reductions to those that never verify the connectedness of reactants. For example, the polymerization (2) is a reaction of this type. It turns out that the WatsonCrick complementary base pairing may be deﬁned in κ−d−u−i .
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Our analysis also takes into account a diﬀerent axis. In [5] a new reaction rule has been introduced, called exchange. According to this reaction, the interaction between two molecules may ﬂip a bond from one to the other. For example, the reader may consider the case where a thief molecule T may connect to a third site of the monomer A and steals the polymer connected to the site l of A: T (t + s), A(h)T (tx + s), A(hx ) T (t + s), A(hx + ly )T (tx + sy ), A(hx + l) x
(4) (5)
(reaction 5 is an example of bond ﬂipping). Bond ﬂipping allows us to model other interesting DNA systems, such as those based on branch migration used to create, for instance, a nanoscale biped walking along a DNA strand [6]. The calculi including bond ﬂipping are made evident with the superscript +bf . Finally, we consider also a more liberal form of ﬂipping, called free ﬂipping (see Figure 3), in which ﬂipping can occur also between two unbound molecules. With free ﬂipping, the thief molecule T can steal the polymer to a monomer without previously connecting to it: T (s), A(ly ) T (sy ), A(l)
(6)
For all of the 14 dialects of κ we investigate three problems: the Reachability Problem (RP), the Simple Coverability Problem (SCP) and the Coverability Problem (CP). The RP is the decision problem associated to the existence of a derivation (simulation) from an initial solution to a target. As shown in [7,8,9], this problem is of high relevance for validation of formal models of biological systems. The SCP is the decision problem associated to the existence of a derivation from an initial solution to a target with given components, regardless of their multiplicity. In other words the shape of complexes in the target solution is taken from a set ﬁxed a priori. SCP is a generalization of the decision problem associated to the static analysis considered in [10]. Finally, CP is the decision problem associated to the existence of a derivation from an initial solution to a target that contains given components. Our results about the (un)decidability of RP, SCP, and CP in the κ lattice are illustrated in Figure 2. The undecidability results are proved by modelling Turing complete formalisms in the calculi, while the decidability results are proved by reduction to decidable properties in ﬁnite state systems or Petrinets. As far as the undecidability results are concerned, the most surprising one is the undecidability of CP in κ−d −u . We prove that this very poor fragment of κ – in which molecules have no state and bonds can be neither destroyed nor ﬂipped – is powerful enough to encode Two Counter Machines [11], a Turing complete formalism. It is also interesting to observe that this result about κ−d −u relies on the possibility to test at least the presence of bonds. In fact, κ−d−u−i is no longer Turing complete because CP is decidable for this fragment (CP allows one to test whether a certain complex, for instance representing the termination of a computation, can be produced). While the dialects that include κ−d −u are Turing complete, many of them retain decidable SCP and/or RP properties. These
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κ+f f κ+f f −n
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Fig. 2. The κ lattice and the (un)decidability of RP, SCP, CP
facts, apparently contrasting with Turing universality of the calculi, are consequences of the following monotonic properties: reactions cannot decrease either (i) the total number of molecules in the solution or (ii) the size of the complexes in the solution. In the calculi satisfying the form of monotonicity (i) we show that it is possible to compute an upperbound to the number of molecules in the solutions of interest for the analysis of RP. In this way, we reduce our analysis to a ﬁnite state system. For the calculi satisfying the form of monotonicity (ii) we show that it is possible to compute an upperbound to the size of the complexes in the solutions of interest for the analysis of SCP. In this case, even if it is not possible to reduce to a ﬁnite state system (because there is no upperbound to the number of instances of the complexes in the solutions of interest), we can reduce to Petrinets in which reachability and coverability are decidable. The paper is organised as follows: Section 2 recalls κ, its fragments and the needed terminology. Section 3 discusses the separation results between the fragments of κ. Section 4 discusses related contributions in literature. Section 5 concludes with few ﬁnal remarks.
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Preliminaries
This section introduces κ and its dialects, together with the terminology that is necessary in the sequel. κcalculi. Two countable sets of species A, B, C, . . ., and of bonds x, y, z, . . . are assumed. Species are sorted according to the number of sites, , and ﬁelds h, i, j, . . . they possess. Sites may be either bound to other sites or unbound, i.e. not connected to other sites. The conﬁguration of sites are deﬁned by partial maps, called interfaces and ranged over by σ, ρ, . . . . The interfaces associate to sites either a bond or a special empty value ε, which models the fact that the site is unbound. For instance, if A is a species with three sites, (2 → x; 3 → ε) is one of its interfaces. This map is written 2x + 3 (the ε is always omitted). We notice that this σ does not deﬁne the state of the site 1, which may be bound or not. Such (proper) partial maps are used in reaction rules in order to abstract from sites that do not play any role in the reactions (similar for evaluations, see below). In the following, when we write σ + σ we assume that the domains of σ and σ are disjoint. The functions dom(·) and ran(·) return the domain and the range of a function. Fields represent the internal state of a species. The values of ﬁelds are also deﬁned by partial maps, called evaluations, ranged over by u, v, . . . . For instance, if A is a species with three ﬁelds, {1 → 5; 2 → 0; 3 → 4}, shortened into 15 + 20 + 34 , is a possible evaluation. We assume there are ﬁnitely many internal states, that is every ﬁeld is mapped into a ﬁnite set of values. As for interfaces, u + v, we implicitly assume that the domains of u and v are disjoint. Definition 1. A molecule A[u](σ) is a term where u and σ are a total evaluation and a total interface of A. Solutions, ranged over by S, T , . . . , are deﬁned by: S ::= A[u](σ)  S, S. Bonds in solutions occur at most twice; in case bonds occur exactly twice the solution is proper. A presolution is a sequence of terms A[u](σ) where u and σ are partial functions and bonds occur at most twice. A presolution is proper if (similarly as before) bonds occur exactly twice. The set of bonds in S is denoted bonds(S). In the rest of the paper the composition operator “,” is assumed to be associative, so (S, S ), S is equal to S, (S , S ) (therefore parentheses will be always omitted). Let σ ≤ σ if dom(σ) = dom(σ ) and, for every i, if σ(i) = ε then σ(i) = σ (i) (the two interfaces may diﬀer on sites mapped to the empty value ε by σ as σ may map such sites to bonds). Reactions have the shape L R, where L and R are presolutions called reactants and products, respectively. The general shape of reactions is deﬁned in the next deﬁnition. Following [5], we extend the deﬁnition of [12] with exchange reactions, thus the calculus is an extension of the κcalculus.1 1
Another diﬀerence with [12] is that we allow newly produced molecules unbound from existing ones.
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Fig. 3. Bond ﬂipping and free ﬂipping
Definition 2. Reactions of the κ+ﬀ calculus – the κ calculus with free ﬂipping rules – are either creations C, or destructions D, or exchanges E. The format of creations is A1 [u1 ](σ1 ), . . . , An [un ](σn ) A1 [u1 ](σ1 ), . . . , An [un ](σn ), B1 [v1 ](φ1 ), . . . , Bk [vk ](φk ) where, for every i, dom(ui ) = dom(ui ), σi ≤ σi , and vi and φi are total. Reactants and products are proper. The format of destructions is A1 [u1 ](σ1 ), . . . , An [un ](σn ) Ai1 [ui1 ](σi1 ), . . . , Aim [uim ](σim ) where i1 , . . . , im is an ordered sequence in [1 . . . n], for every ij , dom(uij) = dom(uij ), σij ≥ σij , and if ij ∈ / {i1 , . . . , im } then σij is total. Reactants and products are proper. The format of exchanges is A[u](ax + σ), B [v](b + ρ) A[u ](a + σ), B [v ](bx + ρ) where ran(σ) = ran(ρ). Creations may change state, produce new bonds between two unbound sites, or synthesise new molecules. Destructions behave the other way around. Exchanges are reminiscent of the π calculus because they deﬁne a migration of a bond from one reactant to the other. We distinguish two types of exchanges: the one occurring between connected molecules, called (connected) bond ﬂipping, and the one occurring between disconnected molecules, called free (bond) ﬂipping. These are illustrated below: The operational semantics of κ+ﬀ calculus uses the following two deﬁnitions: – the structural equivalence between solutions, denoted ≡, is the least one satisfying (we remind that solutions are already quotiented by associativity of “,”):
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• S, T ≡ T, S; • S ≡ T if there exists a renaming ı on bonds such that S = ı(T ). – A1 [u1 + u1 ](σ1 ◦ ı + σ1 ), . . . , An [un + un ](σn ◦ ı + σn ) is an (ı, u1 , · · · , un , σ1 , · · · , σn ) instance of A1 [u1 ](σ1 ), . . . , An [un ](σn ) if ı is a renaming on bonds and the maps uj + uj and σj ◦ ı + σj are total with respect to the species Aj . Definition 3. The reduction relation of the κ+ﬀ calculus, written →, is the least one satisfying the rules: – let L R be a reaction of κ+ﬀ , S be an (ı, u , σ )instance of L, and T be an (ı, u , σ )instance of R. Then S → T ; – let S → T and (bonds(T ) \ bonds(S)) ∩ bonds(R) = ∅, then S, R → T, R; – let S ≡ S , S → T , and T ≡ T , then S → T . The κ+ﬀ calculus groups several subcalculi that have in turn simpler formats of rules. We have already depicted in Figure 2 the fragments we study. We move from κ+ﬀ along two diﬀerent axes: 1. along one axis we restrict the shape of destructions rules: – the superscript −n: we restrict destructions by letting im = n (i.e. forbidding cancellations of molecules), – the superscript −d: we remove destructions, – the superscript −d−u: we remove destructions and consider species with no ﬁelds, – the superscript −d − u − i: we remove destructions, ﬁelds and we restrict creations and exchanges such that no bond occurs in the lefthand side except the ﬂipping one, 2. along the other axis we restrict the capabilities of the exchange rule: – the superscript +bf : we restrict exchanges by allowing bondﬂipping only, – no superscript: we remove exchanges. Some of the combinations are empty. For example, a calculus without checks of bonds and with cancellation of bonds is meaningless as, in order to remove one bond, it is necessary to test its presence ﬁrst. The reader may refer to the introduction for formalisations of relevant biological systems written in these calculi. Decision problems for qualitative analysis. A ﬁrst basic qualitative property is whether a solution eventually produces “something relevant” or not. Clearly this “something relevant” can be deﬁned in a variety of ways. In this paper we consider its formalisation in terms of reachability and coverability, two standard properties which have been extensively investigated in many concurrent formalisms. Few preliminary notions are required. Definition 4 (Complex). Given a proper solution, a complex is a subsolution that is connected (there is a path of bonds connecting every pair of molecules therein) and proper. Two complexes in a solution are equal if they are structurally equivalent.
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Let S(S) be the set of diﬀerent complexes in S; let also →∗ be the transitive and reﬂexive closure of →. Definition 5. RP: the reachability problem of T from a proper solution S checks the existence of R such that S →∗ R and R ≡ T ; SCP: the simple coverability problem of T from a proper solution S checks the existence of R such that S →∗ R and S(R) = S(T ) and R ≡ T, T , for some T ; CP: the coverability problem of T from a proper solution S checks the existence of R such that S →∗ R and R ≡ T, T , for some T .
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(Un)Decidability Results for κ Dialects
In this section we study the (un)decidability of RP, SCP, and CP in the κ lattice of Figure 2. The overall results represented in that ﬁgure are the consequences of theorems that we detail in the remainder of this section. For each decidability region – one for RP, one for SCP, and one for CP – we prove that the corresponding property is decidable in the top language of the region and undecidable in the bottom language(s) among those not included in the region. We separate the presentation of our results in two subsections, the ﬁrst one is devoted to decidability, the latter to undecidability. 3.1
Decidability Results
The proofs of decidability follow by reduction to decidable problems in either ﬁnite state systems or Place/Transition Petri nets (P/T nets). These nets are an interesting inﬁnite state model for the representation and analysis of parallel processes because they retain several decidability problems, such as reachability or coverability [13]. We recall here the basic notation, for a full description of this computational model see [14]. Definition 6. A P/T net is a tuple N = (S, T, F, m0 ), where S and T are ﬁnite sets, called places and transitions, respectively, such that S ∩ T = ∅. A ﬁnite multiset over the set S of places is called a marking, and m0 is the initial marking. F is the transition function associating to each transition t two markings called the preset and the postset of t. The marking m of a P/T net can be modiﬁed by means of transitions ﬁring: a transition with preset m and postset m can ﬁre if m ⊆ m; upon transition ﬁring the new marking of the net becomes (m \ m ) ∪ m where \ and ∪ are the diﬀerence and union operators for multisets, respectively. Our ﬁrst positive result is for the κ+ﬀ −n fragment. Theorem 1. RP is decidable in κ+ﬀ −n . Proof. We reduce RP to the reachability problem in a ﬁnite state system. Let R be a set of κ+ﬀ −n reactions and let S and T be two proper solutions. We
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notice that, in order for S →∗ T , all intermediary solutions traversed by the computation must have a number of molecules which is less or equal to the number nT of molecules in T . This is because in κ+ﬀ −n it is not possible to delete molecules. Let A be the set of species occurring either in S or in a rule of R. Let also setT (A) be the set of (proper) solutions with a number of molecules less than nT . This set is ﬁnite upto structural equivalence because the number of sites and ﬁelds of species is ﬁnite, the values of ﬁelds is ﬁnite, and the possible combinations of bonds is ﬁnite, as well. By mapping every solution R to its canonical representative in the structural equivalence class, called [R], we can build a ﬁnite state system FSST such that, by Deﬁnition 3, given two solutions in setT (A), R → R if and only if [R] → [R ]. We conclude the proof by observing that S →∗ T if and only if [S] →∗ [T ], and this latter property is decidable in FSST . The proof technique adopted above cannot be used to prove the decidability of the SCP problem for a given target T in κ+bf −d . As a matter of fact, SCP allows one to specify only lower bounds, and no upper bounds, to the number of instances of complexes (thus also of the molecules) in the target solution. For this reason ﬁnite state systems are not suﬃciently expressive to model the computations of interest. Nevertheless, we can move to P/T nets because it is possible to compute a ﬁnite set SETT (A) containing the kinds of complexes to be considered in the reachability analysis. This set turns out to be ﬁnite since in κ+bf −d the size of one complex can never decrease and the size of the biggest complex in T ﬁxes an upper bound to the size of the complexes in SETT (A). The idea is then to map each complex in SETT (A) into one place, and deﬁne transitions according to the considered reactions. Hence we have the following theorem: Theorem 2. SCP is decidable for κ+bf −d . The P/T net described above cannot be used to prove the decidability of the CP problem for a given target T in κ+bf −d . In fact, according to CP , the target T indicates only part of the complexes to be reached. Thus, the reached solution that contains the target complexes, could also contain other complexes of size greater than the size mT of the biggest complex in T . Nevertheless, since in κ−d−i bond names cannot be tested in the reactants of a reaction, we can remove from the P/T net representation of those complexes the structure of their bonds, and thus consider only the states and the free sites of their molecules. More precisely, the P/T net described above is now extended with places A[u](σ) (for every species A, every evaluation u, and with partial functions σ mapping every site to ε) used to represent the molecules in complexes of size greater than mT . Due to the ﬁniteness of species, evaluations, and sites we have that this additional set of places is ﬁnite. Moreover the set of transitions is straightforwardly extended to cope with the new places. Hence it is possible to prove the following: Theorem 3. CP is decidable in κ−d−i .
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Undecidability Results
Our undecidability results follow by reduction to undecidable problems such as the halting problem for Two Counter Machines (2CMs), which is a Turing equivalent formalism. A 2CM [11] is a machine with two registers R1 and R2 holding arbitrary large natural numbers and a program P consisting of a ﬁnite sequence of numbered instructions of the following type: – j : Succ(Ri ): increments Ri and goes to the instruction j + 1; – j : DecJump(Ri , l): if the content of Ri is not zero, then decreases it by 1 and goes to the instruction j + 1, otherwise jumps to the instruction l; – j : Halt: stops the computation and returns the value in the register R1 . A state of the machine is given by a tuple (j, v1 , v2 ) where i indicates the next instruction to execute (the program counter) and v1 and v2 are the contents of the two registers. The user has to provide the initial state of the machine. In the rest of the paper, we consider 2CMs in which registers are initially set to zero and where the instruction 0 is Halt. Our ﬁrst negative result is for reachability of a solution in κ. Theorem 4. RP is undecidable in κ. Proof. We reduce the termination problem for 2CMs to RP. Let M be a 2CM with n instructions. To encode it in κ we use ﬁve species: 1. P is the program counter; it retains one ﬁeld with values in [0, . . . , n] and no site; 2. Z1 and Z2 , both with one site, represent the value 0; 3. R1 and R2 , both with two sites, represent the unity to be added to or removed from registries. Let j, l ∈ [0..n] and let i ∈ {1, 2}. The encoding [[·]]κ is deﬁned as follows: P [1j ], Zi (1) P [1j+1 ], Zi (1x ), Ri (1x + 2) [[j : Succ(Ri )]]κ = P [1j ], Ri (2) P [1j+1 ], Ri (2x ), Ri (1x + 2) [[j : DecJump(Ri , l)]]κ
[[j : Halt]]κ =
⎧ ⎨ P [1j ], Zi (1) P [1l ], Zi (1) = P [1j ], Zi (1x ), Ri (1x + 2) P [1j+1 ], Zi (1) ⎩ P [1j ], Ri (2x ), Ri (1x + 2) P [1j+1 ], Ri (2) ⎧ ⎨ P [1j ], Z1 (1), Z2 (1) P [10 ], Z1 (1), Z2 (1) P [1j ], Zi (1x ), Ri (1x + 2) P [1j ], Zi (1) ⎩ P [1j ], Ri (2x ), Ri (1x + 2) P [1j ], Ri (2)
It turns out that the 2CM halts if and only if the solution P [10 ], Z1 (1), Z2 (1) The encoding of 2CMs described above does not apply to κ−n because in this dialect molecules cannot be removed. Nevertheless, we can rephrase the decrement operation of the encoding above by breaking the link between the two last molecules Ri (or the molecules Zi and Ri in case the register holds 1). Hence we have the following theorem:
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Theorem 5. SCP is undecidable in κ−n . We observe that, without using ﬁelds and destructions, as in κ−d −u , it is not possible to reuse the encoding scheme above. Nevertheless, using only creations we can model registers with grids containing two classes of molecules: the molecules of the ﬁrst class represent units in the register, while those of the second class are used to replace units during decrement instructions. Given the register Ri holding n, the corresponding grid contains in the topmost row n molecules of the ﬁrst class. More precisely, the encoding of the increment increases the topmost row of the grid with a molecule of the ﬁrst class. The encoding of the DecJump instruction is more complex: The idea is to copy the topmost row of the grid replacing, if possible, one molecule of the ﬁrst class with one of the second class. If this replacement occurs the subsequent instruction is activated, otherwise a jump is performed. Finally, the encoding of the Halt instruction simply produces the Halt molecule. Given this construction it follows that: Theorem 6. CP is undecidable in κ−d −u . The previous encoding of 2CMs does not allow us to prove the undecidability of SCP in κ+ﬀ −d −u because the exact structure of the grids representing the two registers at the end of the computation is unknown as it depends on the number of increment and decrement instructions that are executed. Nevertheless, in κ+ﬀ −d −u we can use free ﬂipping to “destruct”, at the end of the computation, the grids obtaining an unknown amount of complexes with a known structure. More precisely, we extend the previous construction in such a way that the molecule Halt triggers the following computation: one molecule is produced for each end of each bond in the grids, and all those ends are then passed to such new molecule. Thus we can state the following theorem: Theorem 7. SCP is undecidable in κ+ﬀ −d −u .
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Related Work
In this section we discuss some related works by ﬁrst focusing on formal models speciﬁcally proposed for describing biological systems and then considering more generally the ﬁelds of term/graph rewriting and process calculi. As we said in the Introduction, the closest work to this contribution is [10] where a syntactic restriction entailing a form of SCP is proposed. This restriction – κ with local rule sets – is orthogonal to the ones proposed in this paper. It does not cover the reachability analysis of ﬁnite structures with recurrent patterns, such as ﬁnite polymers. In these cases, the analysis in [10] yields an overapproximation of the reachable complexes. Apart from κ, the literature reports several proposals for describing (and reasoning on) biological systems, which use a variety of formal tools, including process calculi, term/graph rewriting, (temporal) logic, and rule based languages. However, the expressive power of most of these formalisms is the one of Petri
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nets. Therefore, the decidability of reachability and coverability problems is an immediate consequence of the corresponding results on Petri nets. Formalisms whose expressive power is similar to κ, miss results analogous to those contained in this paper. For example, the biochemical abstract machine Biocham [15,16] is a rulebased model similar to κ. However reactions are constrained to specify completely the reagent solution, unlike κ where reactions partially specify reactants and products. It is worth noticing that the Biocham constraint do not allow ﬁnite descriptions of rules creating polymers of arbitrary length. As a consequence, when considering purely qualitative aspects, i.e. removing kinetic quantities, the Biocham can be reduced to a classical Petri net [15]. Another rulebased model for describing and analysing biological processes is Pathway Logic [17,18]. This model is based on rewrite logic, which allows to describe biological entities and their relations at diﬀerent levels of abstractions and granularity by using elements of an algebraic data type (to describe states) and rewrite rules (to describe transitions between states and therefore behaviours). Even though Pathway Logic models of biological processes are developed in Maude system, which is Turing complete, yet the analysis of biological systems uses the, so called, Pathway Logic Assistant for representing models in terms of Petri Nets [18]. Therefore, also in this case, the relevant decidability results derive from the analogous results on Petri nets. This is the case also for the model used in [19]. A diﬀerent model, based on graph transformation has been proposed by Blinov et al. [20]. However, in this case, the relevant properties (e.g. membership of a given species in a reaction network) are semidecidable and we are not aware of suitable restrictions on the general model that ensure decidability for some of them. We have not ﬁnd results regarding the ﬁelds of term/graph rewriting and process calculi, from which we can immediately derive the ones obtained here. In particular, for term rewriting systems, the reductions to Petri net reachability can be applied to decide reachability for associativecommutative ground term rewriting (AC) [21] and for Process Rewrite Systems (PRS) [22]. However, AC and PRS are more expressive than Petri nets, but strictly less expressive than Turing machines [22]. On the other hand our positive results are given for fragments of κ that are Turingcomplete. As such, the set of derivatives of a κ solution may not be a regular set of terms. Thus, decision procedures based on tree automata like those proposed in fragments of nonground term rewriting [23,24,25,26] cannot be applied to the κlattice. Decidability results for reachability in process calculi like Mobile Ambients, Boxed Ambients, and Bioambients are given in [27,28,29,30,31]. These results are obtained for fragments (or for weak semantics) that ensure the monotonicity of the generated ambient structures. In addition they consider process calculi (Mobile/Boxed/Bio Ambients) which operate on treelike structures and without fresh name generation. This contrasts with the dialect of κ of Figure 2, that operate on (possibly cyclic) graphstructures and admit dynamic creation of new names (bonds).
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Concerning Graph Rewriting Systems (GRS) there exist folk theorems about reachability that state its undecidability in fullﬂedged GRS and its decidability for GRS in which rules do not add new nodes. We are not aware of (un)decidability results for decision problems like reachability and coverability in graph rewriting systems with features similar to those considered in our κlattice. The only speciﬁc results we are aware of are those given for reachability in contextfree graph grammars [32] and for coverability in GRS that are wellstructured with respect to the graph minor relation [33]. However, we consider here more general rules than those of contextfree graph grammars. Furthermore, we do not see how to apply the decision procedure proposed in [33] to languages in the κlattice that, in general, do not enjoy strict compatibility with respect to the graph minor ordering.
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Conclusions
We have investigated three decidability problems for several κ dialects. These problems allow one to check whether, starting from a given initial solution, a sequence of reactions described in the κ formalism produces a solution having some speciﬁc features. Hence our results, summarized in Figure 2, can be seen as a ﬁrst step in the direction of qualitative analysis of κ calculus. Besides presenting techniques for qualitative analysis, we also characterise the computational power of κlike biologically inspired models. In this respect, the main result is that we can remove bond and molecule destruction and the internal state of molecules from κ without losing Turing completeness On the contrary, if we remove the possibility to test the presence of one bond in a reaction, the calculus is no longer Turing universal. Our work can be extended along at least two lines. First, several other fragments of κ can be considered for a similar investigation. Notably nanoκ that admits at most two reactants. In particular, our encoding of a 2CM into κ−d−i uses ternary (at the left hand side) rules and we conjecture that a 2CM cannot be encoded faithfully into κ−d−i with binary rules only. Second, there are several other interesting properties to investigate, for example a form of coverability where one admits complexes strictly larger than the original ones. In this perspective, we plan to exploit the theory of well structured transition systems [34] as done in [33] to prove decidability of coverability w.r.t. the graph minor relation in classes of graph rewriting systems.
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Approximation of Event Probabilities in Noisy Cellular Processes Fr´ed´eric Didier1 , Thomas A. Henzinger1, Maria Mateescu1 , and Verena Wolf1,2 1
2
EPFL, Switzerland Saarland University, Germany
Abstract. Molecular noise, which arises from the randomness of the discrete events in the cell, significantly influences fundamental biological processes. Discretestate continuoustime stochastic models (CTMC) can be used to describe such effects, but the calculation of the probabilities of certain events is computationally expensive. We present a comparison of two analysis approaches for CTMC. On one hand, we estimate the probabilities of interest using repeated Gillespie simulation and determine the statistical accuracy that we obtain. On the other hand, we apply a numerical reachability analysis that approximates the probability distributions of the system at several time instances. We use examples of cellular processes to demonstrate the superiority of the reachability analysis if accurate results are required.
1 Introduction The traditional approach for a dynamical model of cellular reaction networks is based on the assumption that the concentrations of the chemical species change continuously and deterministically in time. During the last decade, however, stochastic models with discrete state spaces have seen growing interest [25,30,36,7,29,38,40,43]. The reason is that they take into account the effects of molecular noise in the cell. Molecular noise has a significant influence on important processes such as gene expression [19,3,27,24,6,39], decisions of the cell fate [1,23,22], and circadian oscillations [11,2,12]. An appropriate modeling approach for systems that are subject to molecular noise is a discretestate continuoustime Markov process, also called continuoustime Markov chain (CTMC). This is particularly evident in the presence of intrinsic noise arising from random microscopic events in the cell, such as the location of molecules or the order of the reactions. As opposed to continuous models, the discretestate stochastic model is able to capture the discreteness of the random events in the cell. The evolution of such a CTMC is given by a master equation that is derived according to Gillespie’s theory of stochastic chemical kinetics [10]. Since the state space grows exponentially in the number of involved chemical species, the state space of the CTMC is large, which renders its analysis difficult. Moreover, the discrete structure becomes
This research was supported in part by the Swiss National Science Foundation under grant 205321111840 and by the Excellence Cluster on Multimodal Computing and Interaction.
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even larger when the number of molecules in the system grows. If the populations of certain chemical species are large, their effect on the system’s variance is small and they can be approximated assuming a continuous deterministic change. For species with small populations, however, a continuous approximation is not appropriate and other approximation techniques are necessary to reduce the computational effort of the analysis. Besides the computation of cumulative measures such as expectations and variances of the populations of certain chemical species, the computation of event probabilities is important for several reasons. First, cellular process may decide probabilistically between several possibilities, e.g., in the case of developmental switches [14,1,30]. In order to verify, falsify, or refine the mathematical model based on experimental data, the likelihood for each of these possibilities has to be calculated. But also full distributions are of interest, such as the distribution of switching delays [24], the distribution of the time of DNA replication initiation at different origins [28], and the distribution of gene expression products [42]. Finally, many parameter estimation methods require the computation of the posterior distribution because means and variances do not provide enough information to calibrate parameters [16]. Two different families of computational approaches have been proposed and used to estimate event probabilities and approximate probability distributions. The first kind of approach is based on numerical simulation, i.e., the generation of many sample trajectories (or simulation runs) of the system. The second kind of approach is based on numerical reachability analysis, i.e., the propagation of the probability mass through the state space. The former approach is known as Gillespie simulation [9], in which pseudorandom numbers are used to simulate molecular noise. Measures of interest are obtained via statistical output analysis. The main advantage of simulation is that it is easy to implement and the generation of trajectories is not limited by the size of the state space. Moreover, the precision level of the method can be easily adjusted by performing more or fewer simulation runs. For the computation of the probability of certain events, however, simulative approaches become computationally expensive, because a large number of runs have to be carried out to bound the statistical error appropriately. For estimating event probabilities, a higher precision level is necessary than for estimating cumulative measures such as expectations, and simulation becomes expensive because doubling the precision requires four times more simulation runs. In contrast, approaches based on a numerical reachability analysis approximate probability distributions of the CTMC. As opposed to a statistical estimation of probabilities, which yields an indirect solution, the master equation is numerically solved by integrating the system’s behavior over time. Standard numerical techniques are impractical for many systems because of the enormous size of the state space. Recently, however, more sophisticated numerical approximation methods have been proposed, which solve the system in an iterative fashion and consider only subsets of the state space during any given time interval [17,26,5]. They are significantly more efficient than global analysis because they use localization optimizations (such as “sliding windows”) and dynamic adaptation (“onthefly” generation of windows). These methods efficiently compute the probability distribution of large CTMC at several time instances up to a small approximation error. They can also be used for infinitestate systems.
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In this paper, we evaluate and compare the performance of the two different approaches for the computation of probabilities of certain events, i.e., the statistical estimation using simulation and the approximation using a numerical reachability analysis. For the latter we use a particular algorithm as a representative of the whole family of numerical analysis algorithms, because we have found it to perform best. Similar to the slidingwindow method [17], our algorithm performs a sequence of local analysis steps on dynamically constructed abstractions of the system. The main improvement over the slidingwindow method is that our algorithm is based on adaptive uniformization [41], which allows us to consider arbitrary sets of significant states, i.e., they may be located at different parts of the state space and are not restricted to a specific window shape. Moreover, adaptive uniformization is more robust if the system under study is stiff, i.e., if the chemical reactions occur at time scales that differ by several orders of magnitude. In contrast to [17], here, for the first time, we perform a systematic experimental performance comparison of a numerical reachability analysis with simulation. The first example that we consider is the transcription regulation of a repressor protein in bacteriophage λ [13], where we approximate the probability distribution at several time instances. In the second example, which is a gene expression network [39], we compute the distribution of the time until the number of produced proteins exceeds a certain threshold. In both examples the number of states reachable from the initial state is infinite. The number of chemical species is 6 and 2, and the number of chemical reactions is 10 and 4, respectively. We compare the running time of our numerical reachability analysis to that of the simulative approach for both examples, for different precision levels. Our results show that numerical approximation based on reachability analysis is superior to statistical estimation based on repeated simulation, especially if we increase the desired precision level. For instance, the numerical approximation of the first example needs 39 minutes for a total approximation error of 2 × 10−5 , which distributes among all states. Simulation requires more than six hours if the statistical error of a single event is to be bounded by 10−5 and more than sixty hours for 10−6 .
2 Stochastic Model According to the theory of stochastic chemical reaction kinetics, a continuoustime Markov chain (CTMC) can be derived from a set of biochemical reactions [18,10]. This discretestate model has a regular structure, which gives rise to a functional description in terms of transition class models (TCMs) [33]. TCMs naturally represent coupled chemical reactions as each chemical reaction corresponds to a transition class. They provide, however, a more general description than a set of chemical reactions. 2.1 Transition Class Models Consider a dynamical system with a finite number of discrete state variables such as the number of instances of some chemical species in a reaction volume. Assume that these variables change at discrete points in time. A transition class provides a rule for these changes and a function for the calculation of the statedependent transition rate at which a state change occurs. Let S be a countable set of states.
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Definition 1. A transition class C is a triple (G, u, α) such that (i) the guard G ⊂ S is a subset of S, (ii) u : G → S is an injective update function with u(x) = x for all x ∈ G, (iii) α : G → R>0 is a rate function. A transition class model (TCM) M = (y, {C1 , . . . , Ck }) consists of an initial state y ∈ S and a finite set of transition classes C1 , . . . , Ck . The set G contains all states x in which a transition of type C is possible and u(x) is the target state of the transition. The probability of the Ctransition depends on the transition rate α(x) in the way explained below. In practice, we can usually express G by a finite number of constraints on the state variables, and u and α by elementary arithmetic functions. Thus, a TCM provides a finite description of a (possibly infinitestate) system. Before we show how a CTMC is derived from a TCM, we present some examples of TCMs that describe biochemical reaction networks. Biochemical Reaction Networks. We consider a fixed reaction volume with n different chemical species that is spatially homogeneous and in thermal equilibrium. Then, the state space of the system is given by S = Nn0 . We assume that molecules collide randomly and that collisions may lead to chemical reactions. For a given set of chemical reactions, we construct a TCM such that each transition class corresponds to a reaction and the associated propensity function is given by the rate function α. Example 1. We consider a simple transition class model for transcription of a gene into messenger RNA (mRNA), and subsequent translation of the latter into proteins [39]. This reaction network involves three chemical species, namely, gene, mRNA, and protein. As only a single copy of the gene exists, a state of the system is uniquely determined by the number of mRNA and protein molecules. Therefore, S = N20 and a state is a pair (xR , xP ) ∈ S. We assume that initially there are no mRNA molecules and no proteins in the system, i.e., y = (0, 0). The following four types of reactions occur in the system, namely ∅ → mRNA, mRNA → mRNA + P , mRNA → ∅, and P → ∅. Let i ∈ {1, . . . , 4} and let ci > 0 be a constant. Transition class Ci = (Gi , ui , αi ) describes the ith reaction type. – We describe gene transcription by transition class C1 , which increases the number of mRNA molecules by 1. Thus, u1 (xR , xP ) = (xR + 1, xP ). This transition class is possible in all states, i.e., G1 = S. Transcription happens at the constant rate α1 (xR , xP ) = c1 , as only one reactant molecule (the gene) is available. – We represent the translation of mRNA into protein by C2 . A C2 transition is only possible if there is at least one mRNA molecule in the system. We set G2 = {(xR , xP ) ∈ S  xR > 0} and u2 (xR , xP ) = (xR , xP + 1). Note that in this case mRNA is a reactant that is not consumed. The translation rate depends linearly on the number of mRNA molecules. Therefore, α2 (xR , xP ) = c2 · xR . – Degradation is modeled by C3 and C4 . Hence, G3 = G2 , G4 = {(xR , xP ) ∈ S  xP > 0}, u3 (xR , xP ) = (xR − 1, xP ), and u4 (xR , xP ) = (xR , xP − 1). We set α3 (xR , xP ) = c3 · xR and α4 (xR , xP ) = c4 · xP .
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2.2 Chemical Master Equation A transition class model M = (y, {C1 , . . . , Ck }) represents a timehomogeneous, discretestate Markov process {X(t)}t≥0 , that is, a CTMC with state space S. The jth entry of the random vector X(t) = (X1 (t), . . . , Xn (t)) represents the value of the jth state variable. Let Cm = (Gm , um , αm ), 1 ≤ m ≤ k, and assume that at time t ≥ 0 the process is in state x ∈ Gm . The probability of a transition of type Cm occurring in the next infinitesimal time interval [t, t + τ ), τ > 0 is given by P r(X(t + τ ) = um (x)  X(t) = x) = αm (x) · τ. Since y is the initial state of M we have P r(X(0) = y) = 1, and for x ∈ S we define the probability that X is in state x at time t by p(t) (x) = P r (X(t) = x  X(0) = y) . Recall that um is injective. To simplify our presentation, we define the set Hm as the set of all states x for which u−1 m (x) is defined, that is, that can be reached by a transition of type Cm . The chemical master equation describes the behavior of X by the differential equation [18] ∂p(t) (x) ∂t
=
m:x∈Hm
(t) −1 αm (u−1 m (x)) · p (um (x)) −
m:x∈Gm
αm (x) · p(t) (x) .
(1)
Unbounded Range. For realistic systems, the state space of the Markov chain is extremely large, because its size grows exponentially in the number of involved chemical species. Moreover, if upper bounds on the state variables cannot derived from certain conservation laws, their range is assumed to be infinite although in practice the number of molecules is bounded. Then from the infinite structure, we can compute bounds that are kept with a very high probability. Even though every state in the infinite state space has a nonzero probability, certain attracting regions force most of the probability mass to remain within a finite range. Example 2. In Ex. 1, the degradation rates α3 (x) and α4 (x) grow linearly in the state variables. Thus, the higher the number of mRNA or protein molecules the more likely is their degradation. Depending on the rate constants c1 , . . . , c4 , the system becomes “stable” in different regions. As time approaches infinity, the main part of the probability mass will be close to a region where production and degradation of molecules cancel each other out. Below, we discuss in general under which conditions the system approaches such a stable distribution. Holding Times and Jump Probabilities. A Markov chain {X(t)}t≥0 defined in the way above is a stable and conservative jump process [4]. Thus, there exists a sequence ˆ of jump times {τ (n)}n≥0 and a sequence {X(n)} n≥0 of visited states such that ˆ τ (0) = 0 < τ (1) < τ (2) < . . . and X(t) = X(n) if τ (n) ≤ t < τ (n + 1). ˆ The distribution of the nth holding time τ (n + 1)−τ (n) under the condition X(n) = x is negative exponentially distributed with parameter λ(x) = m:x∈Gm αm (x), also called exit rate of state x.
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If the sum of all holding times is finite with positive probability, the Markov chain is said to explode and the limiting distribution does not exist. Explosive Markov chains are not of interest for the application area of this work since in this case the system “gets lost at infinity”. It is possible to check if the Markov chain does not explode by using Reuter’s Criterion [4]. For the remainder of our presentation we assume that the rate functions αm are such that the Markov chain does not explode. ˆ Assume that the nth state of the Markov chain is x, that is, X(n) = x. If at least one transition class is enabled in x, the successor state is um (x) for some m with x ∈ Gm . The probability of successor um (x) is given by ˆ + 1) = um (x)  X(n) ˆ P r(X(n = x) =
αm (x) λ(x) .
The holding times and the jump probabilities play an important role for the simulation of the Markov chain, which is used to estimate the probability of a certain events.
3 Statistical Estimation of Probabilities In this section we shortly review the basic steps that have to be carried out to estimate the probability of a certain measurable event using stochastic simulation. Throughout this section, we will denote this event by A and its probability by γ. For the analysis of biological systems, the events of interest may be the marginal distributions or even the joint distributions of certain chemical species. For instance, A may have the form Xj (t) = k, that is, the number of type j molecules is k. Estimates are obtained in two steps. In the first step, a certain number of simulation runs of the Markov chain have to be generated, and in the second step, the results of the simulation runs are analyzed. 3.1 Trajectory Generation A realization of the Markov chain, also called trajectory or run, is the random sequence of states visited by the process. If trajectories are produced by a computer, pseudorandom numbers are used to artificially generate randomness [20]. The basic steps of producing a single trajectory that starts in the initial state y at time 0 are as follows: 1. Initialize time t = 0 and state x = y. 2. Generate the holding time h, i.e., a sample of a random variable being exponentially distributed with parameter −λ(x). 3. Generate the successor state, i.e., a sample m of a discrete random variable Z that has probability distribution P (Z = m) = αm (x)/λ(x). 4. Set t = t + h, x = um (x) and go to Step 2 if t < T . In Step 2, we generate the holding time of the current state x. Pseudorandom number generators usually draw from a uniform distribution. Thus, for a given random sample r1 that is uniformly distributed on (0, 1), we calculate an exponentially distributed sample by using the inverse transform method. More precisely, we compute the inverse r1 − ln λ(x) of the cumulative distribution function of the exponential distribution. In Step 3,
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the same idea is used to decide, which reaction occurs next. inverse of the cumuThe i lative distribution function of Z is given by m = min{i : j=1 αj (x) > r2 · λ(x)}, where r2 is again a random sample that is uniformly distributed on (0, 1). In the final step, the current time and the current state are updated. The simulation is terminated if the time horizon T of interest is reached and continued otherwise. 3.2 Output Analysis The problem of estimating the probability γ of the event A can be reformulated as estimating the expectation of the random variable χA with 1 if ω ∈ A, χA (ω) = 0 if ω ∈ A, where ω is a trajectory. The expectation E[χA ] equals γ, since E[χA ] = 1 · P r(χA = 1) + 0 · P r(χA = 0) = γ. Therefore, we can resort to the standard estimation procedure for expectations. Assume that N is the number of runs that have been carried out and Y1 , . . . , YN are independent and identically distributed as χA . Thus, from the ith run we get a realization of Yi by checking if A has occurred or not. It is important to point out that we have to guarantee the independence of the Yi ’s. This implies that we generate 1 N independent trajectories of the Markov chain, each time with a different N initial seed 1 ¯ for the pseudorandom number generator. The sample mean Y = N i=1 Yi is then an unbiased and consistent estimator [20] for E[χA ]. The former means that E[Y¯ ] = E[χA ] and the latter refers to the fact that as N increases the estimator Y¯ becomes closer to γ. Note that Y¯ is equal to the relative frequency of the event A. Let σ 2 = V AR[χA ] be the variance of χA . We evaluate the quality of the estimator Y¯ by applying the central limit theorem, which states that Y¯ will approximately have a Normal distribution with mean E[χA ] = γ and variance σ 2 /N . Hence, for large N the random variable Y¯ − γ Z= σ 2 /N has a standard Normal distribution, that is, the mean is zero and the variance is one. Knowing the distribution of Z enables us reason about the difference Y¯ − γ. Let β ∈ [0, 1] be the confidence level and z ∈ R+ such that β = P r(Z ≤ z). Then Y¯ −γ ¯ − γ ≤ z σ 2 /N . √ β = P r(Z ≤ z) = P r ≤ z = P r  Y 2 σ /N
¯ 2 We estimate σ 2 with the sample covariance S 2 = N1−1 N i=1 (Yi − Y ) , which is an unbiased estimator for σ 2 . Then, for large N and a large number of realizations of the confidence interval
Y¯ − z S 2 /N , Y¯ + z S 2 /N , (2) β is the fraction of intervals that cover γ. It therefore measures the quality of the estimator Y¯ . 1
The seed of a pseudorandom number generator is an initial value, on which the sequence of generated numbers depend [20].
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For a practical application, two further remarks are important. Firstly, we usually choose β ∈ {0.95, 0.99} and the corresponding value of z can be found in the table of the standard Normal distribution. Let Φ be the cumulative distribution function of the standard Normal distribution. Then, using that the Normal distribution is symmetric, 1+β −1 1+β Φ(z) = P r(Z ≤ z) = 1 − 1−β = ⇐⇒ z = Φ . 2 2 2 Secondly, both, Y¯ and S 2 can be computed efficiently N N if during the trajectory generation the realizations of the two sums i=1 Yi and i=1 Yi2 are calculated, since it can be easily shown that S2 =
N
Yi2 N −1 i=1
−
N
(
i=1 Yi ) (N −1)N
2
.
Thus, if r ∈ {0, . . . , N } is the number of times event A occurred during the N simula−r) tion runs, Y¯ = r/N and S 2 = Nr(N (N −1) . If the interval in Eq. 2 is large relative to Y¯ the quality of the estimator is poor and more simulation runs have to be carried out. For our experimental results in Section 5, we fixed the relative width of the interval to be 0.2 (which means that we have a relative error of at most 0.1) and chose confidence level β = 0.95. Thus, z ≈ 1.96 and we can determine the number of necessary runs by bounding the relative width √ z· S 2 /N z2 S 2 S2 2· ≤ 0.2 =⇒ 0.01 γ γ 2 ≤ N =⇒ 384 · γ 2 ≤ N Assume now that we want to estimate the probability of events that occur at least with probability γ. Using the fact that σ 2 = VAR[χA ] = γ(1 − γ) and replacing S 2 by σ 2 yields N ≥ 384 · 1−γ γ [32]. For instance, the sufficient number of runs to guarantee that probabilities, having at least the order of magnitude of 10−5 , are estimated with a relative error of at most 0.1 and a confidence of 95% is N = 38, 000, 000.
4 Numerical Reachability Analysis Instead of indirectly approximating probabilities with statistical estimation procedures, we can use a numerical reachability analysis to solve Eq. 1. An efficient solution by applying standard numerical methods is not possible, since for realistic systems the state space of the system is extremely large. An efficient approximation is, however, possible as long as the total number of involved molecules is a manageable number. We describe a method that is based on a discretization of the process and numerically approximates the probabilities p(t) (x) at certain time instances. Adaptive Uniformization. We discretize the system using adaptive uniformization, which has been introduced by van Moorsel [41] as a variant of standard uniformization [31,34,44,15,35]. Numerical methods based on uniformization have the advantage that they are numerically stable and often more efficient than other methods [37]. We inductively define a sequence S0 , S1 , . . . of subsets of the state space S of the CTMC {X(t)}t≥0 , as well as a sequence p0 , p1 , . . . of functions such that pk : S → [0, 1] for k = 0, 1, . . .. Recall that y is the initial state. We define S0 = {y}, p0 (y) = 1
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λ1
λ2
1
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λ3
2
3
...
Fig. 1. The birth process of the adaptive uniformization procedure
and p0 (x) = 0 if x = y. For k = 1, 2, . . ., we inductively define Sk as follows. We choose a positive uniformization rate λk ≥ maxx∈Sk λx and set Sk+1 = {x ∈ S  ∃x ∈ Sk : pk (x) · qk (x, x ) > 0},
(3)
where, for x ∈ S, ⎧ ⎪ ⎪ m:um (x)=x αm (x)/λk if x = x , ∃m : um (x) = x , ⎨ if x = x , ∃m : um (x) = x , qk (x, x ) = 0 ⎪ ⎪ ⎩ 1 − x ∈S:x =x qk (x, x ) if x = x .
(4)
For x ∈ Sk+1 we set pk+1 (x ) = x ∈Sk pk (x)·qk (x, x ) and pk+1 (x) = 0 if x ∈ Sk . The value pk (x) is the probability of reaching state x after k steps in a discretetime Markov chain {Y (k)}k∈N with transition probabilities P r(Y (k + 1) = x  Y (k) = x) = qk (x, x ) and initial distribution P r(Y (0) = y) = 1. We can reconstruct p(t) (x) by considering an additional process that relates steps with time. Let {B(t)}t≥0 be a birth process with birth rates λ0 , λ1 , . . ., that is, B has a chain structure as illustrated in Fig. 1 and starts initially in state 0 with probability one. In [41], van Moorsel has proved that the original CTMC {X(t)}t≥0 can be constructed from B and Y by setting Y (B(t)) = X(t) if B does not explode. Since Y and B are independent, the state probability p(t) (x) of the original CTMC can be expressed as p(t) (x) =
∞
P r(Y (k) = x) · P r(B(t) = k) =
k=0
∞
pk (x) · P r(B(t) = k).
(5)
k=0
Note that in Eq. 5, there are no negative summands involved. Moreover, pk can be computed inductively. Lower and upper summation bounds L and U can be obtained such that for each state x the truncation error p(t) (x) −
U
pk (x) · P r(B(t) = k) =
k=L
0≤k 500 h −13 140969 10 67 h 22 min
10 10−6
> 3 × 109 > 3 × 108
15 min 41 sec
1 × 10−3
101078 10−12
10−5
> 3 × 107
−3
−11
−4
10
> 3 × 106
10−3
> 3 × 105
6 min 33 sec
7 × 10
67540 10
3 min 12 sec
4 × 10−2
40373 10−10
6 h 44 min 40 min 4 min
Table 2. Comparison of the running times for the gene expression example numerical approximation running time total approx. error Sk  4.2 sec 3.6 sec
Gillespie simulation Δ
5 × 10−6
9816 10−12
−5
−11
5 × 10
−4
8719 10
−10
running time single event error # runs > 500 h
10−7
> 3 × 109
> 50 h
−6
> 3 × 108
−5
10
> 3 × 107
10−4
> 3 × 106
−3
> 3 × 105
3.0 sec
5 × 10
7516 10
2.4 sec
4 × 10−3
6265 10−9 30 min 18 sec
1.9 sec
−2
4 × 10
−8
4939 10
5 h 3 min 3 min sec
10
10
error of insignificant states and, thus, distributes among much more states than only those in Sk . Phage λ Model. The Phage λ model involves 6 different species and 10 reactions. Thus, a state is a vector x = (x1 , x2 , x3 , x4 , x5 , x6 ) ∈ N60 . The transition classes Ci = (Gi , ui , αi ), 1 ≤ i ≤ 10 are given as follows [13]. – Production of proteins: G1 = {x ∈ N60  x3 > 0}, u1 (x) = (x1 + 1, x2 , x3 , x4 , x5 , x6 ), α1 (x) = c1 x3 . – Degradation of proteins: G2 = {x ∈ N60  x1 > 0}, u2 (x) = (x1 − 1, x2 , x3 , x4 , x5 , x6 ), α2 (x) = c2 x1 . – Production of mRNA: G3 = {x ∈ N60  x5 > 0}, u3 (x) = (x1 , x2 , x3 + 1, x4 , x5 , x6 ), α3 (x) = c3 x5 . – Degradation of mRNA: G4 = {x ∈ N60  x3 > 0}, u4 (x) = (x1 , x2 , x3 − 1, x4 , x5 , x6 ), α4 (x) = c4 x3 . – First dimer binding at operator site: G5 = {x ∈ N60  x2 , x4 > 0}, u5 (x) = (x1 , x2 − 1, x3 , x4 − 1, x5 + 1, x6 ), α5 (x) = c5 x2 x4 .the simulation results are still subject to the statistical errors since the true values may not be covered by the confidence interval (compare Section 3.2).
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1 0.8
0.007 0.006
0.6 P
0.005 0.004
0.4
P 0.003 0.2
0.002 0.001 0
0 4000
20 15
Dimers
10 5 2
4
6
8
10
12
14
Monomers
Fig. 2. Probability distribution of monomers and dimers in the phage λ model
16
18
5000
6000
7000
8000
9000
10000
t
Fig. 3. Cumulative probability distribution of the time until the number of proteins reaches 500 for the first time in the gene expression example
– First dimer unbinding: G6 = {x ∈ N60  x5 > 0}, u6 (x) = (x1 , x2 + 1, x3 , x4 + 1, x5 − 1, x6 ), α6 (x) = c6 x5 . – Second dimer binding at operator site: G7 = {x ∈ N60  x2 , x5 > 0}, u7 (x) = (x1 , x2 − 1, x3 , x4 , x5 − 1, x6 + 1), α7 (x) = c7 x2 x5 . – Second dimer unbinding: G8 = {x ∈ N60  x6 > 0}, u8 (x) = (x1 , x2 + 1, x3 , x4 , x5 + 1, x6 − 1), α8 (x) = c8 x6 . – Dimerization: G9 = {x ∈ N60  x1 > 1}, u9 (x) = (x1 − 2, x2 + 1, x3 , x4 , x5 , x6 ), α9 (x) = c9 x1 (x1 − 1)/2. – Dissociation into monomers: G10 = {x ∈ N60  x2 > 0}, u10 (x) = (x1 + 2, x2 − 1, x3 , x4 , x5 , x6 ), α10 (x) = c10 x2 . For c1 , . . . , c10 , we choose c1 = 0.043, c2 = 0.0007, c3 = 0.0715, c4 = 0.0039, c5 = 1.992647 × 10−2 , c6 = 0.4791, c7 = 1.992647 × 10−4 , c8 = 8.765 × 10−12 , c9 = 8.30269 × 10−2, and c10 = 0.5 (see [13,5]). The initial state of the system is given by y = (2, 6, 0, 2, 0, 0) and the time horizon is t = 300. We approximate the probability distributions of the underlying CTMC at 100 equidistant time instances. Fig. 2 shows a plot of the distribution of dimers and monomers at time instant t = 300. In Table 1, we list the running times of our numerical method as well as the running time of the simulation. The column with header Sk  lists the average number of states in the sets S0 , S1 , . . . and Δ is the threshold in Eq. 7. Gene Expression. For the transition classes of the gene expression example we refer to Ex. 1. For the rate constants, we choose c1 = 0.05, c2 = 0.0058, c3 = 0.0029, and c4 = 10−4 , where c3 and c4 correspond to a halflife of 4 minutes for mRNA and 2 hours for the protein [39]. We compute the probability that at least 500 proteins are in the system at 100 equidistant time instances. Fig 3 shows the cumulative probability distribution of the time until the number of proteins reaches 500 for the first time (note that eventually the threshold of 500 is reached with probability one). In Table 2, we list the results for the gene expression example, where, as above, Sk  denotes the average number of states in the sets S0 , S1 , . . . and Δ is the threshold in Eq. 7.
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Discussion. Even if we consider the total approximation error δ as a rough bound for the single error of each state probability, thus favoring simulation, the speedup factor of the numerical approximation is large, especially if the precision increases. The necessary precision level up to which probability distributions are approximated may depend on the system under study. It is, however, important to note that the occurrence of rare biochemical events can have important effects. For instance, the spontaneous, epigenetic switching rate from the lysogenic state to the lytic state in phage λinfected E. coli is experimentally estimated to be in the order of 10−7 per cell per generation [21].
6 Conclusion We have demonstrated that, for the computation of event probabilities, a numerical reachability analysis provides an efficient alternative to simulationbased methods. Even though simulation is widely used, the advantages of numerical methods increase as more sophisticated techniques become available. They reduce the computational effort, especially if accurate results are desired. Moreover, for the calibration of parameters many instances of the model have to be solved and in this case short running times for a single solution are necessary. Until now we have analyzed examples of intrinsically stochastic systems that have been published in the literature. As future work, we are planning to apply our numerical reachability algorithm in collaboration with experimentalists working on new stochastic models. Moreover, we are planning to combine our numerical method with parameter estimation techniques. Standard numerical reachability analysis methods are inefficient for large state spaces (in the case of high dimension and/or many molecules) and inapplicable for unbounded state spaces, and thus one resorts to simulation. We have demonstrated that certain optimization techniques from computer science  localization, on the fly abstraction  put many examples within the reach of numerical reachability analysis. Indeed, when high accuracy is required these methods outperform simulationbased techniques.
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33. Sandmann, W., Wolf, V.: A computational stochastic modeling formalism for biological networks. Enformatika Transactions on Engineering, Computing and Technology 14, 132–137 (2006) 34. Sandmann, W., Wolf, V.: Computational probability for systems biology. In: Fisher, J. (ed.) FMSB 2008. LNCS (LNBI), vol. 5054, pp. 33–47. Springer, Heidelberg (2008) 35. Sidje, R., Burrage, K., MacNamara, S.: Inexact uniformization method for computing transient distributions of Markov chains. SIAM J. Sci. Comput. 29(6), 2562–2580 (2007) 36. Srivastava, R., You, L., Summers, J., Yin, J.: Stochastic vs. deterministic modeling of intracellular viral kinetics. Journal of Theoretical Biology 218, 309–321 (2002) 37. Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton (1995) 38. Swain, P.S., Elowitz, M.B., Siggia, E.D.: Intrinsic and extrinsic contributions to stochasticity in gene expression. PNAS, USA 99(20), 12795–12800 (2002) 39. Thattai, M., van Oudenaarden, A.: Intrinsic noise in gene regulatory networks. PNAS, USA 98(15), 8614–8619 (2001) 40. Turner, T.E., Schnell, S., Burrage, K.: Stochastic approaches for modelling in vivo reactions. Computational Biology and Chemistry 28, 165–178 (2004) 41. van Moorsel, A., Sanders, W.: Adaptive uniformization. ORSA Communications in Statistics: Stochastic Models 10(3), 619–648 (1994) 42. Warmflash, A., Dinner, A.: Signatures of combinatorial regulation in intrinsic biological noise. PNAS 105(45), 17262–17267 (2008) 43. Wilkinson, D.J.: Stochastic Modelling for Systems Biology. Chapman & Hall, Boca Raton (2006) 44. Zhang, J., Watson, L.T., Cao, Y.: A modified uniformization method for the solution of the chemical master equation. TR0731, Computer Science, Virginia Tech. (2007)
Equivalence and Discretisation in BioPEPA Vashti Galpin1 and Jane Hillston1,2 1 LFCS, School of Informatics, University of Edinburgh Centre for Systems Biology (CSBE), University of Edinburgh
[email protected],
[email protected] 2
Abstract. BioPEPA is a process algebra for modelling biological systems. An important aspect of BioPEPA is the ability it provides to discretise concentrations resulting in a smaller, more manageable state space. The discretisation is based on a step size which determines the size of each discrete level and also the maximum number of levels. This paper considers the relationship between two discretisations of the same BioPEPA model that differ only in the step size and hence the maximum number of levels, by using the idea of equivalence from concurrency and process algebra. We present a novel behavioural semantic equivalence, compression bisimulation, that equates two discretisations of the same model and we show that this equivalence is a congruence with respect to the synchronisation operator.
1 Introduction The use of process algebras for modelling biological systems has become a popular technique [1,2,3,4,5]. Some approaches use the process algebra as originally defined for description of computer systems and in others a process algebra is tailored to be specific to systems biology. One of the latter class is BioPEPA [6] which was developed from the stochastic process algebra PEPA [7] and has been successfully used to model Goldbeter’s model of cyclin oscillation [8,9], the Repressilator [10], genetic networks [6], the MAPK model [11], the Neurospora circadian clock [12] and the gp130/JAK/STAT pathway [13]. This paper investigates a semantic equivalence for BioPEPA. An important aspect of BioPEPA is the ability it provides for the discretisation of concentrations. Instead of working with a “processasmolecule” approach, it uses a “processasspecies” approach whereby a process can either be parameterised by concentration or by a discrete level which is obtained from dividing the concentration into a discrete number of chunks. Typically, there is a fixed step size and each species has a maximum number of levels dependent on its maximum concentration. For a given step size, we call a system with levels a discretisation. BioPEPA distinguishes itself from many other process algebras for modelling biological systems by providing multiple analyses including continuoustime Markov chains (CTMCs), ordinary differential equations (ODEs), stochastic simulation and model checking. By developing a semantic equivalence for BioPEPA, we have a new type of analysis based on behaviour that can be used to compare models. Semantic equivalences are an important technique in process algebra for specifying notions of equivalent behaviour. They equate processes that have the same behaviour, P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 189–204, 2009. c SpringerVerlag Berlin Heidelberg 2009
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and can be divided into qualitative equivalences which only consider the behaviour in terms of which actions can be performed and quantitative equivalences which consider the rates at which actions can happen as part of the behaviour. In this paper we consider a qualitative equivalence. Adding quantitative aspects is ongoing research. Semantic equivalences typically have two important properties – they are equivalence relations (hence the name) and congruence relations. A congruence relation is a relation that is preserved by the operators of the process algebra. For example, if is a binary operator, then an equivalence ≡ is preserved by this operator if P ≡ Q implies that P R ≡ Q R and R P ≡ R Q for every process R. These properties are important when modelling and evaluating the concurrent behaviour of computer systems as they let us substitute like for like thereby exploiting the compositionality provided by a process algebra. This allows for the substitution of a system with a smaller state space or other desirable properties, and make the analysis of the system easier. In applying the idea of a semantic equivalence in systems biology, similar advantages will be gained hence the importance and timeliness of this research. Moreover, since the semantic equivalence we define is based upon what reactions can be observed, it is well suited for biological modelling. In searching for a suitable equivalence, there are at least two approaches that can be taken. One is to consider existing equivalences from the literature. The other is to consider what behaviours we want to consider as identical and to develop an equivalence from this starting point. This is the approach taken here. We have a immediate candidate for what we want to consider the same. For a BioPEPA system, we can consider two different discretisations of that system. Since they both represent the same system, we want their behaviours to be identified (assuming neither have few enough levels to give pathological behaviour). This approach is suitable since semantic equivalences are used to identify the same behaviour in different abstractions of a system, and clearly two discretisations are two abstractions. Starting from this point, we define an equivalence relation over the states of the model that relates states that have the same possible reactions. This equivalence relation defines equivalence classes of states with the same behaviour and from this we can use a classical notion of equivalence, bisimulation, to define our semantic equivalence. This new semantic equivalence, compression bisimulation, has not been chosen randomly but through understanding the differences between discretisations and ensuring that the semantic equivalence has the desirable properties mentioned above. Ensuring a semantic equivalence is an equivalence relation is not hard. Establishing congruence is much harder because stoichiometry coefficients greater than one lead to a complex transition system. Being able to prove congruence played a major role in the selection of compression equivalence. Moreover, since two discretisations of a single species should have the same behaviour under the equivalence, it was necessary to prove this as well. The first result of this paper shows that in the sequential case, a single species, two discretisations are related by compression bisimulation. The second shows that compression bisimulation is a congruence with respect to the cooperation operator. A corollary of this is that in the general case of a BioPEPA system, namely with multiple species, two discretisations are related by compression bisimulation.
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The structure of the document is as follows. First, we introduce BioPEPA, then we present some basic ideas relating to semantic equivalences and then we present compression bisimulation and show it has the desired properties. We then give an example.
2 BioPEPA This section presents an overview of BioPEPA [6]. The main components of a BioPEPA system are the sequential components describing the behaviour of each of the species and the model component describing the interactions between the species and initial amounts. Additionally, a context is defined, including functional rates, compartments, parameters. The syntax of the sequential (species) components is defined as S ::= (α, κ) op S  S + S  C
op ::= ↓  ↑  ⊕   .
In the prefix term (α, κ) op S, α is an action name and can be viewed as the name or label of a reaction, κ is the stoichiometry coefficient of the species and the prefix combinator op represents the role of the element in the reaction. Specifically, ↓ is a reactant, ↑ a product, ⊕ an activator, an inhibitor and a generic modifier. The operator + expresses the choice between possible actions and the constant C is defined def by an equation C = S. The syntax of model components is defined as
P ::= P P  S(x) L The process P Q denotes the synchronisation between components P and Q and L the set L specifies those activities on which the components must synchronise. In the model component S(x), the parameter x ∈ R represents the concentration or level. We work with a constrained set of BioPEPA model components as given by the following definition which specifies a wellformed set of components. We ensure that a species consists of a choice between reactions, and no reaction name is repeated within a species. At the model level, there can only be one species component for each species. Definition 1. A BioPEPA sequential component C is welldefined if it has the form n def def C = (α1 , κ1 ) op1 C + . . . + (αn , κn ) opn C written as C = (αi , κi ) opi C i=1
where αi = αj for i = j. A BioPEPA model component P is welldefined if it has the form def . . . L Cp (xp ), P = C1 (x1 ) L 1
p−1
each Ci is a welldefined sequential component, the elements of each Lj appear in P and if i = j then Ci = Cj . We define a BioPEPA system, consisting of a set of welldefined sequential components, a welldefined model component and context, as follows.
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Definition 2. A BioPEPA system P is a 6tuple V, N , K, F , Comp, P , where V is the set of compartments, N is the set of quantities describing each species, K is the set of parameters, F is the set of functional rates, Comp is the set of welldefined sequential components and P is a welldefined model component. Elements of N have the form C : H = h, N = n, M = m, V = v, unit = u where C is a species name that is defined in Comp, H = h defines the step size, N = n defines the maximum number of levels for C, M = m defines the maximum concentration for C, V = v names the compartment in which C appears and unit = u defines the measurement unit of the concentration.
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The notation T , P will be used for V, N , K, F , Comp, P since the details of the tuple are not relevant here. The model component is defined in terms of concentrations, but can be expressed terms of levels where these are a discretisation of the concentration. In N , for each species, there are three elements H, N and M which represent the step size, the maximum number of levels and the maximum concentration respectively. Their relationship is defined as N = M/H . A species with a concentration then has an associated level in the range 0, 1, . . . , N − 1, N , giving a total of N + 1 possible levels. We call the system obtained in this way a BioPEPA system with levels. We will assume that the step size H is the same for all species to ensure conservation of mass. The operational semantics for BioPEPA systems with levels is given in Tables 1 and 2. In the first table, NS refers to the maximum number of levels for the species S. In the rule coop3, w1 ::w2 represents list concatenation. For the rule Final, rα [w, N , K] = fα [w, N , K]/H ∈ (0, ∞) where fα is the functional rate for the reaction α from F and H is the step size. We do not discuss this or the string w further as the equivalences in this paper only consider the action α and ignore the rest of the transition label. The operational semantics creates three different transition systems. The rules for →c define the capability relation. The rule Final defines the system/stochastic relation − →s which includes the context, and the rate at which the transition takes place appears − together with the action. The rule Enrich defines the systemcapability relation − →sc . This relation is necessary since it contains the context information as well as the detailed information that is captured in the list of strings w. The following definition describes the derivative set for the relation − →sc . In this paper, we work almost exclusively with this relation since it provides the necessary information about the context. Definition 3. The systemcapability derivative set ds(P) is the smallest set such that (α,r) P ∈ ds(P) and if P ∈ ds(P) and P −−−→sc P then P ∈ ds(P). The next definition captures the reactions that are immediately possible with respect to the operational semantics. This means it takes into account the stoichiometry of a reaction as well as the current level of a species. Definition 4. The set of current actions enabled in T , P is defined as A( T , P ) = A(P ) where NS is the maximum number of levels for species component S. A(((α, κ) ↓ S)(l)) = {α} if κ ≤ l ≤ NS otherwise ∅ A(((α, κ) ↑ S)(l)) = {α} if 0 ≤ l ≤ NS − κ otherwise ∅ A(((α, κ) ⊕ S)(l)) = {α} if 0 < l ≤ NS otherwise ∅ A(((α, κ) S)(l)) = {α} if 0 ≤ l ≤ NS otherwise ∅ A(((α, κ) S)(l)) = {α} if 0 ≤ l ≤ NS otherwise ∅ A((S1 + S2 )(l)) = A(S1 (l)) ∪ A(S2 (l)) A(C(l)) = A(S(l)) where C = S A(P1 P2 ) = A(P1 ) \ L ∪ A(P2 ) \ L ∪ (A(P1 ) ∩ A(P2 ) ∩ L) L def
The stoichiometry plays a role in defining the set of current actions. A species definition specifies a set of actions (reactions), but the current action set may be a subset if the current level is insufficient to satisfy the constraints imposed by the stoichiometry.
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Since we are working with BioPEPA systems that vary only in step size and maximum numbers of levels, we require notation to capture this. Given a BioPEPA system P we can define a system where the lowest maximum number of levels for any species is n. As mentioned previously, we assume that H is identical for all components. Definition 5. Let P = V, N , K, F , Comp, P be a BioPEPA system with welldefined P parameterised by concentration. For n ∈ N, the BioPEPA system with levels P n is defined as P n = V, N , K, F , Comp, P where 1. γ = (1/n) · min{m  C : H = h, N = n , M = m, V = v, unit = u ∈ N } 2. C : H = h, N = n , M = m, V = v, unit = u ∈ N ⇒ C : H = γ, N = m/γ , M = m, V = v, unit = u ∈ N def def . . . L Cp (xp ) ⇒ P = . . . L Cp ( xp /γ ) 3. P = C1 (x1 ) C1 ( x1 /γ ) L L 1
1
p−1
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N contains information about each species. The definition above identifies the species with the smallest concentration, determines the new step size that will ensure n levels for that species and then adjusts the other species to use the same step size (to conserve mass) hence modifying N . Since P is a BioPEPA system with species components parameterised by concentration and we wish to work with a system with levels, the initial concentrations in the third point are transformed to initial levels. We will use the notation P n = T n , P to indicate that the lowest number of maximum levels for any species is n and refer to P n as a discretisation of P.
3 Semantic Equivalences In process algebras, a semantic equivalence defines what it means for two models to have the same behaviour. A classical notion of equivalence is that of bisimulation [14]. Definition 6. A binary relation R is a bisimulation if for any (P, Q) ∈ R and for any θ whenever 1. P − → P , there exists Q such that Q − → Q and (P , Q ) ∈ R, and θ θ 2. Q − → Q , there exists P such that P − → P and (P , Q ) ∈ R θ
θ
P and Q are bisimilar, P ∼ Q if (P, Q) ∈ R for some bisimulation R. This leads to the definition ∼ = {R  R a bisimulation} and one can show that ∼ is the largest bisimulation. Moreover, it can also be shown that bisimulation is an equivalence relation therefore it is reflexive, symmetric and transitive. Bisimulation is a finegrained notion of behaviour and equates far fewer models than language/trace equivalence, for example. It requires that related models can match each other’s transitions and that the resultant models also have this property. Consider the labelled transition systems in Figure 3. They generate the same strings/traces but they are not bisimilar because we cannot find anything to match with Q1 . Q2 is not suitable since it only has a b transition and Q2 is not suitable since it only has a c transition. As mentioned in the introduction, we also wish that our new semantic equivalence be a congruence with respect to the language we use.
Equivalence and Discretisation in BioPEPA P1 a
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Fig. 4. Example of discretisations that are not bisimilar
3.1 Compression Bisimulation We now define the new equivalence. As noted in the introduction, our approach here is to consider the systems we want to be equivalent and to work from there. We want our equivalence to be a congruence and we also want to equate discretisations with sufficiently large numbers of levels because this is our starting point. However, having said that, we are still interested in an equivalence that is similar to classical equivalences such as bisimulation. Note that we cannot use bisimulation directly here. This can be def shown by the species component A = (α, 1)↑A. Figure 4 gives the transition system for two different discretisation, one where the maximum level is 2 and the other where it is 3. We can relate T 2 , A(i) and T 3 , A(i) for 0 ≤ i ≤ 2 but we cannot relate
T 3 , A(3) to any of T 2 , A(i). However, although we cannot use bisimulation directly, we are able to use it indirectly over equivalence classes and achieve the goals of congruence and equating discretisations. We now present definitions that allow us to achieve that. We first need to define the equivalence relation that will define the relevant equivalence classes. The current level together with the stoichiometry associated with a reaction determine which reactions can occur, therefore we are interested in grouping together those states of the BioPEPA system for which the same reactions can take place. The collection of enabled reactions becomes our underlying notion of behaviour. This captures the similarities that we see between different discretisations. From a biological point of view, this is sensible because it is an observational notion of equivalence. In light of this, we can define an equivalence relation over BioPEPA systems that depends on A which defines the actions that are currently enabled. Two processes are related if their current action sets are the same. Definition 7. The current action relation H over welldefined BioPEPA systems is defined as H = {(P1 , P2 )  A(P1 ) = A(P2 )}. Proposition 1. H is an equivalence relation.
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Because H is an equivalence relation, it defines equivalence classes of BioPEPA systems which have the same current actions. For a set of BioPEPA systems X , the equivalence classes of X with respect to H is denoted X /H. A can be extended to the equivalence classes in the obvious manner. Hence, for P ∈ H an equivalence class, A(H) = A(P). We are interested in considering the equivalence classes over the derivative set of a given BioPEPA system P because we want to consider the overall behaviour of individual BioPEPA systems and we define PH = ds(P)/H. Since we want to define a bisimulationstyle equivalence we need to define transitions between equivalence classes. The basic idea is that if there is a transition between individual members of two equivalence classes then there is a transition between those equivalence classes. α Definition 8. For H, H ∈ PH , H −→ H if there exists P ∈ H and P ∈ H such (α,w) that P −−−→sc P .
We can then finalise the definition for our new equivalence as follows. We use Definiθ α tion 6 for the definition of ∼ but substitute −→ for all instances of − → and, moreover the relation ∼ is defined between equivalence classes. Definition 9. P and Q are compression bisimilar, P Q, if PH ∼ QH . 3.2 Equivalence and Congruence Results This section details results about the new equivalence. First we need show that it is an equivalence relation. Proposition 2. P Q is an equivalence relation. Proof. This is straightforward because ∼ is an equivalence relation.
Next, we consider the sequential case of two discretisations and show that they are equated by the new equivalence. The sequential case consists of considering a single species and two discretisations. The first theorem of the paper shows that given a single species component and two discretisations, then the two discretisations are compression bisimilar because their induced equivalence classes are bisimilar (in fact, they are isomorphic). First, some notation and various lemmas are required. The complexity of these results is due to the factthat stoichiometry can be larger than one. For a sequential def m BioPEPA component C = i=1 (αi , κi ) opi C, let T↑ = {κi  (αi , κi ) ↑ C appears in the definition of C} T↓ = {κi  (αi , κi ) ↓ C appears in the definition of C} ∪
t↑ = T↑ 
{1  (αi , κi ) ⊕ C appears in the definition of C} AC = {αi  (αi , κi ) opi C appears in the definition of C}
t↓ = T↓ 
km = max{k↓ , k↑ , 1}, k↓ = max(T↓ ), k↑ = max(T↑ ), hence km ≥ k↓ , km ≥ k↑ . The diagram in Figure 5 illustrates the equivalence classes for two discretisations of def C = (α, 2) ↑ C + (β, 3) ↑ C + (γ, 4) ↓ C + (δ, 1) ⊕ C with n = 11 and n = 13. It
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Fig. 5. The equivalence classes of two discretisations of a species component
also demonstrates how the various stoichiometry coefficients result in different equivalent classes. The lemmas that follow prove the properties of the equivalence classes as shown in this diagram. The next lemma establishes that equivalence classes can be ordered which makes them easier to manipulate in later lemmas. The following lemma builds on this and shows that there are a fixed number of equivalence classes if there are sufficient levels and it contributes to the definition of the isomorphism in the first theorem. def m Lemma 1. For a sequential BioPEPA component C = i=1 (αi , κi ) opi C and the n BioPEPA system S n = T n , C, the equivalence classes of SH form a strict order. Proof. The set ds(S n ) contains elements of the form T n , C(l), where l ranges over 0, . . . , n. Each equivalence class is a subsequence of 0, . . . , n because of the side conditions of the prefix rules. Therefore a class can be described by its smallest and largest elements [i, j] for i ≤ j. These intervals do not overlap because the equivalence classes form a partition. Hence for any two equivalence classes [i, j] and [i , j ], either j < i or j < i, and this property defines a strict order over the equivalence classes. def m Lemma 2. For a sequential BioPEPA component C = i=1 (αi , κi ) opi C and the n n n BioPEPA system S = T , C, if n ≥ k↑ + k↓ + 1, then SH has t↑ + t↓ + 1 equivalence classes. Proof. By Lemma 1, a sequence of equivalence classes [i1 , j1 ], [i2 , j2 ], . . . , [it , jt ] parn titioning SH exist. We show that there is an equivalence class [ic , jc ] with A([ic , jc ]) = AC , ic = k↓ and jc = n − k↑ . This is welldefined since n ≥ k↓ + k↑ + 1. Consider l ∈ [ic , jc ]. Any production prefix (α, κ)↑C is enabled since 0 ≤ ic ≤ l ≤ n − k↑ ≤ n − κ. Any reactant prefix (α, κ)↓C is enabled because κ ≤ k↓ ≤ l ≤ n − k↑ ≤ n. Prefixes containing or can always generate transitions. Since l ≥ 1, any prefix of the form (α, κ) ⊕ C is enabled. Hence A([ic , jc ]) = AC and the class cannot be larger. Next we consider the equivalence classes that come before [ic , jc ]. Order the elements of T↓ from smallest to largest, τ1 , τ2 , . . . , τt↓ −1 , τt↓ where τt↓ = k↓ then we have that [i1 , j1 ], [i2 , j2 ], . . . , [ic−1 , jc−1 ] is [0, τ1 − 1], [τ1 , τ2 − 1], . . . , [τt↓ −1 , τt↓ − 1] which gives t↓ equivalence classes. Similarly, we have t↑ equivalence classes corresponding to the elements of T↑ . This means that there are t↓ + t↑ + 1 equivalence classes in total.
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def Corollary 1. Let C = m i=1 (αi , κi ) opi C be a sequential BioPEPA component which has stoichiometry coefficient 1 for all reactant prefixes and product prefixes then any discretisation with n ≥ 3 has three equivalence classes.
The next lemma establishes a lower bound on the size of the equivalence class that is capable of performing all actions, namely the “central” class [ic , jc ] with A([ic , jc ]) = Ac . This class is the only one that differs in cardinality for different discretisations, and it grows in size as the number of levels are increased. The other classes do not differ between different discretisations because they are defined by the same stoichiometry coefficients. The value km is used since knowing that km is a bound on the size of [ic , jc ] is important for a later lemma. def m Lemma 3. Let C = i=1 (αi , κi ) opi C be a sequential BioPEPA component and let S n = T n , C for n ≥ k↑ + km + k↓ . Then [ic , jc ], the equivalence class of S n /H with A[ic , jc ] = AC , has cardinality greater than km .
Proof. It is the case that ic = k↓ and jc = n − k↑ . The cardinality of [ic , jc ] is jc − ic + 1 = n − k↑ − k↓ + 1 ≥ k↑ + km + k↓ − k↑ − k↓ + 1 = km + 1 > km . This implies that the cardinality/width of [ic , jc ] is greater than both k↓ and k↑ . The next lemma relates the equivalence classes obtained for two different values of n by expressing the classes for the larger value in terms of the intervals of the smaller value. def Lemma 4. Let C = m i=1 (αi , κi ) opi C be a sequential BioPEPA component and n let S = T , C. Let n = n + d. Then the equivalence classes of SH are described by the ordered intervals [0, j1 ], . . . , [ic , jc ], . . . , [it−1 , jt−1 ], [it , n] and the equivalence n classes of SH are described by the ordered intervals [0, j1 ], . . . , [ic , jc + d], . . . , [it−1 + d, jt−1 + d], [it + d, n + d] where [ic , jc ] and [ic , jc + d] are the equivalence classes in which all actions of C are possible.
n Proof. The elements of SH are [0, j1 ], . . . , [ic , jc ], . . . , [it−1 , jt−1 ], [it , n] and those of n SH are [0, j1 ], . . . , [ic , jc ], . . . , [it−1 , jt−1 ], [it , n ]. The first t↓ equivalence classes are the same in both cases because they are defined by the same stoichiometric coefficients n so SH is [0, j1 ], . . . , [ic , jc ], . . . , [it−1 , jt−1 ], [it , n ]. Since jc = n−k↑ and jc = n −k↑ , jc = jc + d and this offset is the same throughout the remaining equivalence classes n which are defined by the same stoichiometric coefficients hence SH can be written [0, j1 ], . . . , [ic , jc + d], . . . , [it−1 + d, jt−1 + d], [it + d, n + d].
Finally the most important lemma shows that the same transitions occur between equivalence classes if the numbers of levels are large enough. This contributes to the isomorphism defined in the theorem about sequential BioPEPA systems. def m Lemma 5. Let C = i=1 (αi , κi ) opi C be a sequential BioPEPA component and let n S = T , C. Let E1 , . . . , Et be the ordered equivalence classes of SH and E1 , . . . , Et α n be the ordered equivalence classes of SH . If n ≥ k↓ + km + k↑ then Ep −→ Eq if and α only if Ep −→ Eq .
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Proof. Let n = n + d. Lemma 4 gives notation for the two sequences of equivalence classes and let [ic , jc ] and [ic , jc ] be the classes whose current action set is AC . Conα α sider [ip , jp ] −→ [iq , jq ]. Since A([ip , jp ]) = A([ip , jp ]), [ip , jp ] −→ [i , j ]. We need (α,w) n to show that [i , j ] = [iq , jq ]. There are two transitions T , C(l) −−−→ T n , C(l1 ) (α,w ) with l ∈ [ip , jp ] and l1 ∈ [iq , jp ] and T n , C(l ) −−−−→ T n , C(l1 ) with l ∈ [ip , jp ] and l1 ∈ [i , j ]. So we need to show that l1 ∈ [iq , jq ]. We also need to conα α sider the other direction. If [ip , jp ] −→ [iq , jq ] then [ip , jp ] −→ [i, j] and we need to show that l ∈ [iq , jq ]. Prefixes (α, κ) op C where op ∈ {⊕, , }: A transition does not result in a change of level therefore p = q and [i , j ] = [ip , jp ]. Similarly for the other direction. We now consider the relative positions of p, q and c. The cases (p < c and q > c) and (q < c and p > c) cannot occur due to Lemma 3 as no transition can change the value of l or l sufficiently. Prefixes of the form (α, κ)↑C: Here the level l changes to l + κ and p ≤ q. If q < c then l ∈ [ip , jp ] and l + κ ∈ [iq , jq ] hence choosing l = l leads to l ∈ [ip , jp ] and l + κ ∈ [iq , jq ] as required. The other direction is similar. For q = c and p < c, let l = l then l +κ ∈ [ic , jc +d] since l +κ ∈ [ic , jc ]. For the other direction, l ∈ [ip , jp ] and l +κ ∈ [ic , jc +d]. We need to show that l+κ ∈ [ic , jc ]. The largest value l can take is ic − 1. The width of [ic , jc ] is jc − ic + 1 > km by Lemma 3. Therefore l+κ = l +κ ≤ ic −1+κ ≤ ic −1+km < ic −1+jc −ic +1 ≤ jc . If p > c, then l ∈ [ip , jp ] and l+κ ∈ [iq , jq ], Let l = l+d then l ∈ [ip +d, jp +d] and l + κ ∈ [iq + d, jq + d] = [iq , jq ] as required. For the other direction, let l = l − d. If p = c and q = c let l = l then l + κ ∈ [ic , jc ] implies l + κ ∈ [ic , jc + d]. For the other direction, we need to show that l, l + κ ∈ [ic , jc ]. Choose l = ic and note that l + κ = ic + κ ≤ ic + km < jc + 1 by Lemma 3 therefore l + κ ≤ jc . If p = c and q > c then l ∈ [ic , jc ] and l + κ ∈ [iq , jq ]. To match this, choose l = l + d. For the other direction, l ∈ [ic , jc + d] and l + κ ∈ [iq + d, jq + d]. Let l = l − d, then l ∈ [ic − d, jc ] and l + κ ∈ [iq , jq ]. We need to show l ≥ ic . The smallest value that l + κ can take is jc + d + 1 which implies l + d + κ = l + κ ≥ jc + d + 1. So l ≥ jc − κ + 1 ≥ jc − km + 1 > ic by Lemma 3. Prefixes of the form (α, κ)↓C: The level changes from l to l − κ and p ≥ q. These are similar to the previous case. The following theorem shows that for large enough value of n, two discretisations of a sequential BioPEPA system are compression bisimilar. Theorem 1. Let S = T , C be a welldefined BioPEPA system with the single species def m component C = i=1 (αi , κi ) opi C then S n S n for n, n ≥ k↓ + km + k↑ . Proof. Without loss of generality, assume that n is the maximum number of levels for species C in S n and n is the maximum number of levels for species C in S n . n n n n We will show that SH is isomorphic to SH hence SH ∼ SH and therefore S n S n . n n Let f : SH → SH be defined as f (B) = D if A(B) = A(D). This function is welln n defined by Lemma 2 since SH and SH have the same number of equivalence classes n n and hence for B, B ∈ SH , f (B) = f (B ) implies B = B and for any D ∈ SH , there n exists B ∈ SH such that f (B) = D.
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α α Additionally, define f (B −→ B ) = f (B) −→ f (B ). This is a homomorphism beα n cause it preserves transitions. By Lemma 5, for any D −→ D , there exist B, B ∈ SH α such that f (B) −→ f (B ). Hence f is a isomorphism.
Classically in congruence proofs, there would be a proof for each operator, hence there would be one for each of the prefix operators and then one for the choice operator. We do not need to show that the new semantic equivalence is a congruence with respect to the prefix operators and the choice operator since we work specifically with welldefined model components which give a constrained syntax that restricts how the prefix operators and the choice operator can be used. We next consider a congruence result for the synchronisation operator. In this theorem, the notation [P]H refers to the equivalence class generated by H, the current action relation, that contains the BioPEPA system P. From this it is possible to obtain the result about compression bisimilarity between two different discretisations of a model component. First, we need a lemma and then a property describing the actions that are possible in a synchronisation. The property captures the idea that the actions in L are those that are shared by both components in the synchronisation. Lemma 6. Equality with respect to A is preserved by cooperation. In other words, A( T , P1 Q) = A( T , P2 Q) and L L A( T , P1 ) = A( T , P2 ) ⇒ A( T , Q P ) = A( T , Q P 1 2 ) L L
Definition 10. Given a welldefined BioPEPA system of the form T , P Q, it has L the current action decomposition property if A( T , P1 Q ) = A( T , P Q2 ) 1 2 L L implies A( T , P1 ) = A( T , P2 ) and A( T , Q1 ) = A( T , Q2 ) for all systems
T , P1 Q1 , T , P2 Q2 ∈ ds( T , P Q). L L L Theorem 2. Let T1 , P1 , T2 , P2 , T1 , Q1 and T2 , Q2 be welldefined BioPEPA systems such that T1 , P1 Q1 and T2 , P2 Q2 both have the current action L L decomposition property. If T1 , P1 T2 , P2 and T1 , Q1 T2 , Q2 then it is the case that T1 , P1 Q1 T2 , P2 Q2 . L L Proof. LetR = [ T1 , P1 Q1 ]H , [ T2 , P2 Q2 ]H  [ T1 , P1 ]H ∼ [ T2 , P2 ]H L L and [ T1 , Q1 ]H ∼ [ T2 , Q2 ]H . We want to show that R is a bisimulation. This proof follows the standard technique for this case but requires a few additional steps. We only consider the case of α ∈ L. The other two cases are simpler. Consider a transition from [ T1 , P1 Q1 ]H . This may be generated by a transition L (α,w) from another system in this class, say T1 , P3 Q3 −−−→sc T1 , P3 Q3 . By L L shorter inferences and by the definition of transitions between equivalence classes we α α can infer [ T1 , P3 ]H −→ [ T1 , P3 ]H and [ T1 , Q3 ]H −→ [ T1 , Q3 ]H . From the current action decomposition property, we know that these transitions are α α [ T1 , P1 ]H −→ [ T1 , P3 ]H and [ T1 , Q1 ]H −→ [ T1 , Q3 ]H . α By the definition of R, there exist P2 and Q2 such that [ T2 , P2 ]H −→ [ T2 , P2 ]H α and [ T2 , Q2 ]H −→ [ T2 , Q2 ]H such that [ T1 , P3 ]H ∼ [ T2 , P2 ]H and [ T1 , Q3 ]H ∼ [ T2 , Q2 ]H .
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α β
2210 γ 2301
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α β α β α β
0030 γ
E1
0121 γ
α β, γ α β
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α
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0212 γ
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Fig. 6. Transition system for M C when n = 3 (left) and associated equivalence classes (right) (α,w )
We can then infer T2 , P4 −−−−2→sc T2 , P4 with A( T2 , P4 ) = A( T2 , P2 ) (α,v2 ) and A( T2 , P4 ) = A( T2 , P2 ) and T2 , Q2 −−−− →sc T2 , Q2 with A( T2 , Q4 ) = A( T2 , Q2 ) and A( T2 , Q4 ) = A( T2 , Q2 ). By Lemma 6 and the above equalities, A( T2 , P4 Q4 ) = A( T2 , P2 Q2 ) L L and A( T2 , P4 Q4 ) = A( T2 , P2 Q2 ). L L (α,w) We can infer T2 , P4 Q − − − →
T Q4 which leads to the matching sc 2 , P4 4 L L α transition [ T , P Q ] − → [ T , P Q ] and by definition we know that 2 H 2 H 2 L 2 2 L 2 [ T1 , P3 Q ] , [ T , P Q ] ∈ R. H 2 H 3 2 L 2 L
Corollary 2. Let P = T , P be a welldefined BioPEPA system then P n P n for n, n ≥ k↓ + km + k↑ if the current action decomposition property applies to pairs of subcomponents of P n and P n .
. . . L Cm (lm ) with each Ci a seProof. Since P is welldefined, P = C1 (l1 ) L m−1 1 quential component. By Theorem 1 and n, n > k↓ + km + k↑ , T n , Ci T n , Ci for all i. By repeated applications of Theorem 2, T n , P T n , P . def
4 Example We give an example of discretisations and the associated equivalence classes. Consider the substrateenzymeproduct reactions S +E SE → P +E which can be expressed in BioPEPA as S = (α, 1) ↓ S + (β, 1) ↑ S def
E = (α, 1) ↓ E + (β, 1) ↑ E + (γ, 1) ↑ E def
SE = (α, 1) ↑ SE + (β, 1) ↓ SE + (γ, 1) ↓ SE def
P = (γ, 1) ↑ P def
E(x) {α,β,γ} SE(0) P (0) M C = S(x) {α,β} {γ} def
Figure 6 gives the transition system for M C when the maximum number of levels for all species is three as well as that for the equivalence classes. In the transition system for M C each state is a BioPEPA system and is indicated by its vector representation which describes the level of each species in that system using the vector (S, E, SE, P ) but we omit the brackets and commas in the diagram. Additionally, we have omitted the strings
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α 6610 5520 γ β γ α 6701 5611 β γ 5702
α
α
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β α
4430 γ 4521 β γ α 4612 β γ 4703
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3340 γ 3431 γ 3522 γ 3613 γ 3704
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2250 γ 2341 γ 2432 γ 2523 γ 2614 γ 2705
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1160 γ 1251 γ 1342 γ 1433 γ 1524 γ 1615 γ 1706
α β α β α β α β α β α β α β
0070 γ 0161 γ 0252 γ 0342 γ 0614 γ 0525 γ 0616 γ 0707
Fig. 7. The transition system for M C when n = 7
w from the diagram for reasons of space. The shading shows the different equivalence classes in both diagrams. Figure 7 gives the system when the maximum number of levels is seven and the equivalence classes are also shown. This demonstrates how the two discretisations are related by the equivalence classes given in Figure 6.
5 Related Work The use of process algebras for modelling systems biology has multiplied rapidly since the first paper advocated the use of the πcalculus [15]. Approaches include the κcalculus [2], stochastic πcalculus [3,1], Betabinders [4] and BioAmbients [5]. Most of these approaches use stochastic simulation as their analysis tool, and few approaches have considered the use of semantic equivalences. Weak bisimulation is shown to be a congruence for the bioκcalculus as is a context bisimulation which allows for the modelling of cell interaction [16]. Observational equivalence has been used to show that CCS specifications of elements of lactose operon regulation have the same behaviour as more detailed models [17]. In an example of biological modelling using hybrid systems, bisimulation is used to quotient the state space with respect to a subset of variables as a technique for state space reduction [18]. Bisimulation has also been used in the comparison of ambientstyle models and membranestyle models [19] and the comparison of a termrewriting calculus and a simple brane calculus [20].
6 Conclusions and Further Research This paper has presented a new semantic equivalence for BioPEPA called compression bisimulation and shown that it is a congruence and it identifies different discretisations of the same system. A first step for further work is to find a syntactic characterisation of the BioPEPA systems that exhibit the current action decomposition property so that those systems with this property can be easily identified. Since reactions have unique names in BioPEPA, an extension that would be useful is to allow for a relation over names to relate different reactions in different models. We would also like to extend the equivalence to be quantitative and take into account reaction rates but it is not immediately obvious
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how to do this. Finally, we wish to apply the equivalence to various biological models and to other formalisms using discretisation. Acknowledgements. Vashti Galpin is supported by EPSRC Grant EP/E031439/1. Jane Hillston is supported by EPSRC ARF EP/C543696/01. The Centre for Systems Biology at Edinburgh is a Centre for Integrative Systems Biology (CISB) funded by the BBSRC and EPSRC in 2006. We thank Maria Luisa Guerriero for her comments.
References 1. Blossey, R., Cardelli, L., Phillips, A.: A compositional approach to the stochastic dynamics of gene networks. In: Priami, C., Cardelli, L., Emmott, S. (eds.) Transactions on Computational Systems Biology IV. LNCS (LNBI), vol. 3939, pp. 99–122. Springer, Heidelberg (2006) 2. Danos, V., Laneve, C.: Formal molecular biology. Theoretical Computer Science 325, 69– 110 (2004) 3. Priami, C., Regev, A., Shapiro, E., Silverman, W.: Application of a stochastic namepassing calculus to representation and simulation of molecular processes. Information Processing Letters 80, 25–31 (2001) 4. Priami, C., Quaglia, P.: Beta binders for biological interactions. In: Danos, V., Sch¨achter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 20–33. Springer, Heidelberg (2005) 5. Regev, A., Panina, E., Silverman, W., Cardelli, L., Shapiro, E.: BioAmbients: an abstraction for biological compartments. Theoretical Computer Science 325, 141–167 (2004) 6. Ciocchetta, F., Hillston, J.: BioPEPA: a framework for the modelling and analysis of biological systems. Theoretical Computer Science (in press) 7. Hillston, J.: A compositional approach to performance modelling. CUP (1996) 8. Ciocchetta, F., Hillston, J.: Calculi for biological systems. In: Formal Methods for Computational Systems Biology. In: SFM 2008. LNCS, vol. 5016, pp. 265–312. Springer, Heidelberg (2008) 9. Ciocchetta, F.: BioPEPA with SBMLlike events. In: Back, R.J., Petre, I. (eds.) Proceedings of the Workshop on Computational Models for Cell Processes, pp. 11–22 (2008) 10. Ciocchetta, F., Hillston, J.: BioPEPA: a framework for the modelling and analysis of biological systems. Technical Report EDIINFRR1231, School of Informatics, University of Edinburgh (2008) 11. Ellavarason, K.: An automatic mapping from the Systems Biology Markup Language to the BioPEPA process algebra. Master’s thesis, University of Trento (2008) 12. Akman, O.E., Ciocchetta, F., Degasperi, A., Guerriero, M.L.: Modelling biological clocks with BioPEPA: stochasticity and robustness for the Neurospora Crassa circadian network. To appear in Proceedings of CMSB 2009 (2009) 13. Guerriero, M.L.: Qualitative and quantitative analysis of a BioPEPA model of the gp130/JAK/STAT signalling pathway. To appear in TCSB 14. Milner, R.: Communication and concurrency. Prentice Hall, Englewood Cliffs (1989) 15. Regev, A., Shapiro, E.: Cellular abstractions: Cells as computation. Nature 419, 343 (2002) 16. Laneve, C., Tarissan, F.: A simple calculus for proteins and cells. Theoretical Computer Science 404, 127–141 (2008) 17. Pinto, M.C., Foss, L., Mombach, J.C.M., Ribeiro, L.: Modelling, property verification and behavioural equivalence of lactose operon regulation. Computers in Biology and Medicine 37, 134–148 (2007) 18. Antoniotti, M., Piazza, C., Policriti, A., Simeoni, M., Mishra, B.: Taming the complexity of biochemical models through bisimulation and collapsing: theory and practice. Theoretical Computer Science 325, 45–67 (2004)
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19. Ciobanu, G., Aman, B.: On the relationship between membranes and ambients. BioSystems (91), 515–530 (2008) 20. Barbuti, R., MaggioloSchettini, A., Milazzo, P., Troina, A.: Bisimulation in calculi modelling membranes. Formal Aspects of Computing 20, 351–377 (2008)
Improved Parameter Estimation for Completely Observed Ordinary Diﬀerential Equations with Application to Biological Systems Peter Gennemark1,2 and Dag Wedelin3 1
3
University of G¨ oteborg, SE412 96 G¨ oteborg, Sweden 2 Uppsala University, SE751 06 Uppsala, Sweden
[email protected] Chalmers University of Technology, SE412 96 G¨ oteborg, Sweden
[email protected] Abstract. We consider parameter estimation in ordinary diﬀerential equations (ODEs) from completely observed systems, and describe an improved version of our previously reported heuristic algorithm (IET Syst. Biol., 2007). Basically, in that method, estimation based on decomposing the problem to simulation of one ODE, is followed by estimation based on simulation of all ODEs of the system. The main algorithmic improvement compared to the original version, is that we decompose not only to single ODEs, but also to arbitrary subsets of ODEs, as a complementary intermediate step. The subsets are selected based on an analysis of the interaction between the variables and possible common parameters. We evaluate our algorithm on a number of wellknown hard test problems from the biological literature. The results show that our approach is more accurate and considerably faster compared to other reported methods on these problems. Additionally, we ﬁnd that the algorithm scales favourably with problem size. Keywords: ordinary diﬀerential equations, parameter estimation, decomposition. Supplementary material: All problems, solutions, online software and supplementary information are available at www.odeidentification.org.
1
Introduction
This paper considers parameter estimation in ordinary diﬀerential equation (ODE) models, applied to biological systems. Usually, such models are based on reaction kinetics and are nonlinear in both variables and parameters, see Fig. 1 for an example. A general way to estimate the parameters is to deﬁne an optimisation problem in which the objective is to ﬁnd parameter values minimising an error function based on the discrepancy between the observed data and the simulated model. P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 205–217, 2009. c SpringerVerlag Berlin Heidelberg 2009
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Fig. 1. Model of a gene network considered in Moles et al. [1]. M0−3 are metabolites, E1−3 are enzymes, and G1−3 are mRNAs. M0 are M3 are input variables, while all others are dependent variables. To each of the dependent variables, there is a corresponding ODE. The ODE for M1 , containing the parameters kcat1 , Km1 , Km2 , kcat2 , Km3 , and Km4 , is shown for illustration. The model contains 36 parameters in total.
The evaluation of the error function is usually slow, since it requires the entire model to be simulated for each experiment. Since this has to be repeated many times, the overall method is computationally intensive for realistic problems. We consider the case where some time course information is available for every state variable, and present an improved version of the algorithm of Gennemark and Wedelin [2], together with empirical evaluation on a number of test problems. Motivation for this work comes from several sources. While many parameter estimation algorithms often allow data to be speciﬁed for only a subset of the variables, further work on a base case is generally of interest if this leads to signiﬁcantly faster algorithms for this case. Another speciﬁc motivation is that we are interested in the use of parameter estimation as a subroutine in a more complex algorithm for identifying also the structure of ODE systems, when the mathematical expressions of the ODEs are unknown. Such algorithms have recently become of increased interest because of the combination of largescale experimental techniques, and increased computational power (see [3] and references therein). To be feasible, this inference problem typically relies on complete data sets in which all state variables are observed. For example, one may identify a gene regulatory network from a set of mRNA time proﬁles collected from a microarray experiment. In the majority of methods proposed in this area, a structure search heuristic iteratively proposes diﬀerent ODE structures to try, and for each such model the parameters are estimated. Usually, most of the computation time is spent on parameter estimation, which hence constitutes a highly time critical subroutine. For this reason, research on parameter estimation on completely observed systems has an direct impact on the more general problem of ﬁnding both structure and parameters.
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For the feasibility of the parameter estimation problem see e.g. Ljung [4], who discusses how problems may not be unambiguously solved without additional data or some additional model constraints. Local optimisation methods like GaussNewton can be used to optimise the error function, and, generally, such methods are computationally fast but can fail because of local optima of the objective function [5, 6]. Standard softwares for local methods are publicly available, and benchmark problems to evaluate these are also available, e.g. in the data base EASYFIT [5]. Global methods, on the other hand, may identify the global optimum but to the cost of an increased computational time [1, 7, 8, 9]. For example, one of the simplest possible global methods is to restart local estimation from several randomly chosen starting points. For problems with complete data sets, and where the derivatives can be estimated with relatively high precision, the ODEs can be transformed to a set of algebraic equations by replacing the lefthand sides by estimated slopes from data. Then, the derivative method [10], or rigorous deterministic methods based on interval analysis [11] can be applied. Compared to these two methods, our own algorithm [2] is also dependent on complete data, but not so much on high precision estimates of derivatives.
2
Parameter Estimation
Since the algorithm is based on Gennemark and Wedelin [2], we refer to that reference for details. Here we brieﬂy review the basic principle of the algorithm and then describe our improvements. 2.1
The Original Algorithm
Input to the algorithm consists of the structure of the model (the ODEs), lower and upper bounds for all parameters, and sets of time series experiments with standard deviations, where every variable should be measured in at least one experiment. The solution is a parameter vector k that minimises an error function, which in principle can take any form. In our case we work with the log likelihood of the observed data. By assuming independent and normally distributed measurement errors and disregarding constant terms we can express the loglikelihood for one time series as 1 L(Xˆj  k) = − 2 i
Xj (ti ) − Xˆj (ti ) σj (ti )
2 (1)
ˆ j and σj denotes simuwhere i indexes the measurement points, and where Xj , X lated data, experimental data and standard deviation for variable j, respectively. ˆ k) is deﬁned by summing over all variables and all The total loglikelihood L(X ˆ k). experiments. The error function to minimise is −L(X
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One main idea in Gennemark and Wedelin [2] is to use the complete data set to decompose the parameter estimation to single ODEs. In this way we obtain one small optimisation problem for each ODE. The decomposition is achieved by using interpolated experimental data for the other variables occurring on the righthand side of the ODE, resulting in a decomposed problem for a single variable. Standard parameter estimation routines (local or global) can then be applied on the decomposed problem, and it is solved by applying simulation of the single ODE to evaluate the error function. As an example of the decomposition, to estimate the six parameters of M1 (t) in Fig. 1, interpolated data are used for M0 , M2 E1 and E2 when simulating the ODE of M1 . The advantage of this approach is twofold. First, the small problems can be solved much faster, and second, the overall stability of the estimation increases, since the decomposed problems are constrained by the empirical data and not by results from the algorithm itself. The disadvantage however, is that the precision may be lower than for estimation of the entire model. To successfully take advantage of this potential speed gain, and at the same time achieve high precision, another idea is to use a mix of diﬀerent estimation methods with diﬀerent properties with respect to speed and precision. Thus, the originally proposed algorithm also occasionally applies estimation to the entire problem. Initially, we also use the very rough but fast derivative method [10], to create a starting point. Finally, we can choose between experimental and simulated data given the best parameters hitherto as input for the subproblems as appropriate (see [2] for details). An iterative procedure is obtained as follows: REPEAT 1. Select input for the other variables, to be used in step 2: experimental data or simulated data. 2. FOR each equation DO (a) Make a rough estimate of the parameters in the equation with the derivative method. (b) Improve by estimation for single equation subproblems. 3. Improve by estimation for the entire problem. UNTIL convergence In step 1, it is natural to consider experimental data in the ﬁrst iteration, and simulated data in subsequent iterations. Note that simulated data is only used to estimate variables and derivatives in the decomposition of step 2, not to replace Xˆj in the error function. In order to avoid local minima, both step 2a and 2b can be repeated with several randomly chosen initial parameters. Typically, we use 60 random starting points in step 2a, and then feed the best parameters into step 2b which is run only once. 2.2
The Improved Algorithm
While the original algorithm is very fast and works well for many problems, we have observed two problems that we address in the improved version:
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– For some systems, if estimation for single equation subproblems have low precision and estimation for the entire problem is too slow, it can be diﬃcult to ﬁnd a good mix between the two. – For systems based on traditional kinetic reactions, it is common that the same reaction term occurs in two of the ODEs, one for production and one for consumption. Using estimation for single equations, the parameters of such reactions will hence be estimated twice, leading to nonoptimal use of data that may result in ambiguity and potentially divergent solutions. To address these problems, we propose a generalisation of our algorithm. Instead of restricting the estimation to one ODE (step 2) or to all ODEs (step 3), we allow parameter estimation of the parameters in any submodel, i.e. by considering any subset of the equations. It is intuitively reasonable that one should select submodels with equations that are strongly linked. This can be deﬁned in diﬀerent ways, for example by a known direct interaction, and/or by common parameters. Such links can be speciﬁed by the user, or be inferred automatically. Here we propose two diﬀerent automatic approaches. The ﬁrst approach is intended for metabolic systems. We have found it useful to deﬁne one submodel for every variable by starting from every single ODE, and ﬁnd those other equations that have reactions (and thus also parameters) in common with this ODE. A shared parameter is usually involved in a reaction term that is consumed in one ODE and produced in another. Submodels that are empty or contained in a submodel of another equation are ignored. For example, equation 1 may share a reaction with equation 2 and another with equation 3. Hence, these 3 equations form the submodel for variable 1. Now, if equation 2 only shares parameters with equation 1, the submodel of equation 2 is contained in the submodel of equation 1 and can be ignored. The second approach is intended for other types of systems that typically lack shared parameters, e.g. genetic networks. Here we let the submodel of each ODE include the variables which occur on the righthand side of the equation. This new step of parameter estimation based on submodels complements the previous parameter estimation steps. In the original algorithm, it is inserted between step 2 and 3, resulting in our modiﬁed algorithm: REPEAT 1. Select input for the other variables, to be used in step 2 and 3: experimental data or simulated data. 2. FOR each equation DO (a) Make a rough estimate of the parameters in the equation with the derivative method. (b) Improve by estimation for single equation subproblems. 3. Improve by estimation of submodels deﬁned around each single ODE. 4. Improve by estimation for the entire problem. UNTIL convergence
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In addition to these changes to the structure of the algorithm, the implementation has been modiﬁed so that common parameters are also recognised in the ﬁnal step of estimation of the entire problem. The actual implementation has also been improved in numerous ways, compared to the original one.
3
Numerical Evaluation on Benchmark Problems
We evaluate the algorithm by comparing it with previously presented results in the literature. The problems we consider are collected from the literature based on the following requirements: (1) all state variables of the system should be observed, and (2) information on accuracy and/or eﬃciency for other methods solving the problem should be reported. All found problems satisfying these criteria have been included in our test set, see Table 1. For problems with noise, many publications only specify the character of the noise, without explicitly supplying data. In these cases we have reconstructed the problems according to the speciﬁcations. All test problems and our solutions are fully speciﬁed on our web site www.odeidentification.org. On the web site, it is also possible to run these problems or own similar problems online on our server. Table 1. Overview of the problems considered in our study . Problems in the upper part of the table are collected from the literature, while the lower part includes problems that are designed for this study. #var and #param correspond to the number of variables and parameters, respectively. For some of the problems with noisy data, the number of model parameters is complemented with (after the plus sign) the number of data parameters (one initial value for each time series). #exp and #tp correspond to the number of experiments and the number of timepoints for each variable in each experiment, respectively. Measurement noise is added from a normal distribution with mean zero and standard deviation proportional to respective data point. Problem
#var
#param
#exp
#tp
Noise
Reference
pe pe pe pe pe pe pe
3genes1 3genes2f 3genes3f pinene ss cascade1 ss branch4 ss 30genes2f
8 8 8 5 3 4 30
36 36 36 5 14 18 128
16 16 16 1 8 4 20
21 21 21 9 41 20 11
0% 3% 5% ≈5% 0% 0% 10%
[1] [12] [12] [13] [14] [15] [15]
pe pe pe pe pe pe
3genes2 3genes3 ss 30genes2 4genes1 5genes1 6genes1
8 8 30 11 14 17
36+128 36+128 128+600 48 60 72
16 16 20 16 16 16
21 21 11 21 21 21
3% 5% 10% 0% 0% 0%
This This This This This This
work work work work work work
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For problems with names beginning with ’pe ss’ we have used the second approach to select submodels. For all other problems we have used the ﬁrst approach. Since most methods in this ﬁeld are dependent on random numbers, the results may diﬀer between runs. Therefore, we run all our tests several times with diﬀerent random seeds. Computation times are reported as the average computation time for runs with diﬀerent random seeds, scaled to a 1GHz processor for easy comparison to other results (our actual runs were performed on a Pentium IV, 2.13GHz). 3.1
Problems from the Literature
An overview of the results is given in Table 2. We ﬁrst consider a known hard test problem based on the model in Fig. 1, for which common estimation algorithms often fail, see [1]. The main challenge of this problem is the wide search ranges for the parameters and the nonlinearities in the ODEs. For this problem, referred to as pe 3genes1, our proposed algorithm returns a perfect solution (L = 0.00), and the eﬃciency is improved compared to all previously reported results on this problem, see Table 2. The parameter accuracy is in the same order as the chosen precision of the numerical integration routine. Notably, the methods we compare to are considered state of the art in the ﬁeld of global estimation for ODEs, as indicated by the extensive comparison in [1], and by further comparisons in [12, 13]. Furthermore, we consider two variations of this problem with Gaussian noise added to data, pe 3genes2f and pe 3genes3f. Also on these problems our algorithm performs better than previously reported methods with respect to eﬃciency, see Table 2. In this case, no comparison on the solution can be made, since we have not used exactly the same data (same noise level but not the same actual data). The next problem, pe pinene, was originally studied by Box et al. [16], and represents a biochemical system with ﬁve variables. Like the method of RodriguezFernandez et al. [13], our method ﬁnds the optimal solution, but with higher eﬃciency, as indicated in Table 2. We also consider three problems with models deﬁned as Ssystems [17, 18], which are based on approximating kinetic laws with multivariate powerlaw functions. The ﬁrst Ssystem problem pe ss cascade1 was introduced by Voit [18] and applied in Tsai and Wang [14]. The model represents a cascaded pathway with three dependent variables. The second Ssystem problem pe ss branch4 was introduced by Voit [18] and applied in Kutalik et al. [15]. It represents a branched pathway with four dependent variables. Finally, the third Ssystem problem pe ss 30genes2f was introduced by Maki et al. [19] and applied in Kutalik et al. [15]. It represents a genetic network with 30 variables. The main challenge of this problem is the large number of parameters. The results presented in Table 2 indicate that our approach is faster and has higher accuracy compared to the evolutionary optimisation approach used by Tsai and Wang [14]. Moreover, the results indicates that our approach is faster compared to the decomposition approach used in Kutalik et al. [15]. Again, since data
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Table 2. Running times and accuracies for our parameter estimation method. For problems in the upper part of the table results for other methods are available and also reported. Computation time is given as the average of several runs, and scaled to a 1GHz processor. Stability measures the frequency of runs for which the best −L was obtained. Problem
Reference
pe 3genes1
External [1] External [12] Best external This work pe 3genes2f Best external This work pe 3genes3f Best external This work pe pinene Best external This work pe ss cascade1 Best external This work pe ss branch4 Best external This work pe ss 30genes2f Best external This work pe pe pe pe pe pe
3genes2 3genes3 ss 30genes2 4genes1 5genes1 6genes1
This This This This This This
work work work work work work
[13] [12] [12] [13] [14] [15] [15]
Best solution Performance −L Accuracy Time (s) Stability 104 3748
> 104 > 104
150 83
180 97
260 120
1/10 1/10
pe pinene
No Yes
9.936 9.936
9.936 9.936
9.936 9.936
0.2 0.1
0.2 0.1
0.2 0.1
9/10 9/10
pe ss cascade1
No Yes
0.00 0.00
0.00 0.00
0.00 0.00
9 11
11 13
14 10/10 15 10/10
pe ss branch4
No Yes
0.00 0.00
0.00 0.00
0.00 0.00
2.2 0.5
2.2 0.5
2.2 10/10 0.5 10/10
pe ss 30genes2f No Yes
3240 3240
3240 3240
3240 3240
310 190
330 210
360 10/10 240 10/10
pe 3genes2
No Yes
2719 1214
> 104 3815
> 104 > 104
240 88
380 110
520 220
1/10 5/10
pe 3genes3
No Yes
3217 2309
> 104 3188
> 104 > 104
210 70
350 160
530 270
1/10 1/10
pe ss 30genes2
No Yes
3004 3004
3004 3004
3004 3004
1900 860
2000 910
2300 10/10 1200 10/10
pe 4genes1
No Yes
0.00 0.00
0.00 0.00
> 104 > 104
1300 610
1300 630
1600 720
pe 5genes1
No Yes
0.00 0.00
0.00 0.00
0.00 0.00
1700 810
1700 840
1800 10/10 860 10/10
pe 6genes1
No Yes
0.00 0.00
0.00 0.00
> 104 > 104
2100 1100
2200 1100
2800 1200
8/10 7/10
8/10 8/10
A Bayesian Approach to Model Checking Biological Systems Sumit K. Jha1 , Edmund M. Clarke1 , Christopher J. Langmead1,2 , Axel Legay3 , Andr´e Platzer1 , and Paolo Zuliani1 1
2
Computer Science Department, Carnegie Mellon University, USA Lane Center for Computational Biology, Carnegie Mellon University, USA 3 Institut d’Informatique INRIA, Rennes, France
Abstract. Recently, there has been considerable interest in the use of Model Checking for Systems Biology. Unfortunately, the state space of stochastic biological models is often too large for classical Model Checking techniques. For these models, a statistical approach to Model Checking has been shown to be an eﬀective alternative. Extending our earlier work, we present the ﬁrst algorithm for performing statistical Model Checking using Bayesian Sequential Hypothesis Testing. We show that our Bayesian approach outperforms current statistical Model Checking techniques, which rely on tests from Classical (aka Frequentist) statistics, by requiring fewer system simulations. Another advantage of our approach is the ability to incorporate prior Biological knowledge about the model being veriﬁed. We demonstrate our algorithm on a variety of models from the Systems Biology literature and show that it enables faster veriﬁcation than stateoftheart techniques, even when no prior knowledge is available.
1
Introduction
Computational models are increasingly used in the ﬁeld of Systems Biology to examine the dynamics of biological processes (e.g., [8,10,20,30,34,37]). By ‘computational’, we mean discretevariable and continuous or discretetime models [4], where the components of the system interact and evolve by obeying a set of instructions or rules. In contrast to diﬀerential equationbased models, which are also widely used in Systems Biology, computational models can provide insights into the role of stochastic eﬀects over discretepopulations of molecules or cells. Recently, there has been considerable interest in the application of Model
This research was sponsored by the GSRC (University of California) under contract no. SA423679952, National Science Foundation under contracts no. CCF0429120, no. CNS0411152, and no. CCF0541245, Semiconductor Research Corporation under contract no. 2005TJ1366, Air Force (University of Vanderbilt) under contract no. 18727S3, International Collaboration for Advanced Security Technology of the National Science Council, Taiwan, under contract no. 1010717, the U.S. Department of Energy Career Award (DEFG0205ER25696), and a Pittsburgh LifeSciences Greenhouse Young Pioneer Award.
P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 218–234, 2009. c SpringerVerlag Berlin Heidelberg 2009
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Checking [15] as a powerful tool for formally reasoning about the dynamic properties of such models (e.g., [1,6,9,11,14,18,24,38]). This paper presents a new Model Checking algorithm that is wellsuited for verifying properties of very large stochastic models, such as those created and used in Systems Biology. The stochastic nature of most computational models from Systems Biology gives rise to an instance of the Probabilistic Model Checking (PMC) problem [13,15,31]. Suppose M is a stochastic model over a set of states S, s0 is a starting state, φ is a dynamic property expressed as a formula in temporal logic, and θ ∈ [0, 1] is a probability threshold. The PMC problem is: given the 4tuple (M, s0 , φ, θ), to decide algorithmically whether M, s0 = P≥θ (φ). In this paper, property φ is expressed in BLTL  Bounded Linear Temporal Logic [36,35,19]. Given these, PMC algorithms decide whether the model satisﬁes the property with at least probability θ. Existing algorithms for solving the PMC problem fall into one of two categories. The ﬁrst category comprises numerical methods (e.g. [2,3,12,16,31]) which can compute the probability with which the property holds with high precision. Numerical methods are generally only suitable for small systems (≈ 106 to 107 states). In a Biological System, the number of states can easily exceed this limit, which motivates the need for algorithms for solving the PMC problem in an approximate fashion. Approximate methods (e.g., [23,26,39,46]) work by sampling a set of traces from the model. Each trace is then evaluated to determine whether it satisﬁes the property. The number of satisfying traces is used to (approximately) decide whether M, s0 = P≥θ (φ). Approximate PMC methods can be further divided into two subcategories: (i) those that seek to estimate the probability that the property holds and then compare that estimate to θ (e.g., [26,39]), and (ii) those that reduce the PMC problem to a hypothesis testing problem (e.g., [46,47]). That is, deciding between two hypotheses — H0 : P≥θ (φ) versus H1 : P T . Jeﬀreys interprets the value of the Bayes factor as a measure of the evidence in favor of H0 (dually, B1 is the evidence in favor of H1 ). We now show how to compute the Bayes factor. According to Deﬁnition 2, we have to calculate the probability of the observed sample d = (x1 , . . . , xn ) given H0 and H1 . They are given by integrating the joint density h(d·) with respect to the prior g(·), and since we assume that the sample is drawn from iid variables, we have that h(d·) = f (x1 ·) · · · f (xn ·). Therefore, the Bayes factor is the ratio:
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B=
1
P (x1 , . . . , xn H0 ) = θ θ P (x1 , . . . , xn H1 )
f (x1 u) · · · f (xn u) · g(u) du
.
(2)
f (x1 u) · · · f (xn u) · g(u) du
0
We observe that the Bayes factor depends on the data d and on the prior g, so it may be considered a measure of conﬁdence in H0 vs. H1 provided by the data x1 , . . . , xn , and “weighted” by the prior g. Hence, the choice of the threshold Bayes Factor (T ) in Sec. 3.2 also indicates an objective degree of conﬁdence in the accepted hypothesis when the Bayesian Statistical Model Checking algorithm stops. 3.2
Algorithm
Our algorithm is essentially a sequential version of Jeﬀreys’ test. Remember we want to establish whether M = Pθ (φ), where θ ∈ (0, 1) and φ is a BLTL formula. Like all statistical Model Checking algorithms, we assume that it is possible to generate unbiased samples from the model. The algorithm iteratively draws independent and identically distributed sample traces σ1 , σ2 , ..., and checks whether they satisfy φ. As explained above, we can model this procedure as independent sampling from a Bernoulli distribution X of unknown parameter p  the actual probability of the model satisfying φ. At stage n the algorithm has drawn samples x1 , . . . , xn iid like X. It then computes the Bayes factor Bn according to (2), and it stops iﬀ (Bn > T ∨ Bn < T1 ). When this occurs, it will accept H0 iﬀ Bn > T , and will accept H1 iﬀ Bn < T1 . The algorithm is shown below. From (2) we see that the algorithm can incorporate prior knowledge through g, when computing the Bayes factor. Our examples focus on Beta priors which are deﬁned over the (0, 1) interval by the following probability density (for real parameters α, β > 0): ∀u ∈ (0, 1) g(u, α, β) =
1 uα−1 (1 − u)β−1 B(α, β)
where the Beta function B(α, β) is deﬁned as: 1 B(α, β) = tα−1 (1 − t)β−1 dt .
(3)
(4)
0
By varying the parameters α and β, one can approximate other smooth unimodal densities on (0, 1) by a Beta density (e.g., the uniform density over (0, 1) is a Beta with α = β = 1). We also deﬁne the Beta distribution function F(α,β) (u): u u 1 ∀u ∈ (0, 1) F(α,β) (u) = g(t, α, β) dt = tα−1 (1 − t)β−1 dt (5) B(α, β) 0 0 which is just the usual distribution function for a Beta random variable of parameters α, β (i.e., the probability that it takes values less than or equal to u).
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Algorithm 1. Bayesian Statistical Model Checking Require: PBLTL Property Pθ (φ), Threshold T > 1, Prior density g for unknown parameter p n := 0 {number of traces drawn so far} x := 0 {number of traces satisfying φ so far} repeat σ := draw a sample trace of the system (iid) n := n + 1 if σ = φ then x := x + 1 end if Bn := BayesFactor(n, x) {compute according to Equation (2)} until (Bn > T ∨ Bn < T1 ) if (Bn > T ) then return H0 accepted else return H1 accepted end if
The choice of the Beta density is not arbitrary. It is wellknown that the Beta distribution is the conjugate prior to the Bernoulli distribution2 . This relationship gives rise to closedform solutions to the posterior density over θ (i.e., P (θd)), thus avoiding numerical integration when calculating the Bayes factor. Our data (x1 , . . . , xn ) are assumed to be iid samples from a Bernoulli drawn n distribution of unknown parameter p. We write x = i=1 xi for the number of successes in (x1 , . . . , xn ). The prior density g(·) is assumed to be a Beta density with ﬁxed parameters α, β > 0. In [29], we show that the Bayes factor Bn at stage n can be computed in terms of the Beta distribution function: Bn =
1 −1 . F(x+α,n−x+β) (θ)
The Beta distribution function can be computed with high accuracy by standard mathematical libraries (e.g. the GNU Scientiﬁc Library) or software (e.g. Matlab). Hence, the Beta distribution is the appropriate choice for summarizing the prior probability distribution in Statistical Model Checking. We present the following two Theorems: Theorem 1 (Termination). The Bayesian Statistical Model Checking algorithm terminates with probability one, for Beta priors and Bernoulli samples. (See [29] for a proof.) 2
A distribution P (θ) is said to be a conjugate prior for a likelihood function, P (dθ), if the posterior, P (θd) is in the same family of distributions.
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Theorem 2. If the Bayesian Model Checking algorithm terminates after observing n sample traces, an upper bound on the probability of the Type I error is n x I{B(n, x) < 1/T }(x) tmax (1 − tmax )n−x x x=0 n
where tmax is the value of t that maximizes the expression ti (1 − t)n−i deﬁned on [θ, 1], T is the Bayes Factor threshold used in the Bayesian Model Checking algorithm, and I is the indicator function. (See [29] for a proof.) 3.3
Veriﬁcation over General Priors
The use of conjugate priors does not pose restrictions, in practice. It is known that any prior distribution (with or without a density) can be well approximated by a ﬁnite mixture of conjugate priors [17]. Thus, we can approximate an arbitrary prior over (0, 1) by constructing a density G(·) of the form: G(u) =
N
ri · gi (u, αi , βi )
i=1
where N is a positive integer which depends on the level of accuracy required, the gi ’s are Beta densities (of possibly diﬀerent parameters αi , βi ), and the ri ’s are positive reals summing up to 1  this ensures that G is a proper density. For such priors, the computation of the Bayes factor is slightly more complicated. In [29], we show that the Bayes factor at stage n is given by: Bn = N
i=1 ri
N
i=1 ri
· B(x + αi , n − x + βi )
· B(x + αi , n − x + βi ) · F(x+αi ,n−x+βi ) (θ)
−1
where ri = B(αrii,βi ) . Again, we see that the Bayes factor can be computed by means of standard, wellknown numerical methods, thereby simplifying the implementation of the algorithm. Theorem 1 can be extended to handle this case, too [29].
4
Benchmarks
In this section, we analyze the performance of our algorithm on ﬁve benchmark models from the Systems Biology literature. Three of the models are written in the prism Model Checking tool’s speciﬁcation language [27,31], and the remaining two are written in SBML and were obtained from the Matlab Systems Biology Toolbox. The prism Model Checker tool is capable of both symbolic (i.e., exact) Probabilistic Model Checking, and statistical Probabilistic Model Checking. prism’s statistical Probabilistic Model Checking Algorithm implements the algorithm of [26] which uses a ﬁxed sized sampling approach and estimates the true probability as the number of satisfying traces over the number of sampled
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traces. We note that for each of the benchmark sets, we consider models that are too large for symbolic model checking. Our experiments demonstrate two important properties of our algorithm: (i) we show that our algorithm requires fewer traces than either the algorithm of [26] implemented in prism or Wald’s SPRT algorithm  while retaining the same bounds on the frequentist TypeI and TypeII error probabilities. (ii) The performance of both the Wald’s algorithm [42] and our Bayesian Model Checking algorithm degrades as the threshold probability (i.e., θ) in the PBLTL temporal logic formula gets close to the actual probability of the model satisfying the BLTL formula. However, the Bayesian algorithm shows a more graceful degradation compared to Wald’s SPRT approach. 4.1
PRISM Benchmarks
We studied three large PRISM benchmarks which are not well suited for numerical approaches to Probabilistic Model Checking. In our experiments, the Bayesian Model Checking algorithm used uniform priors, and accepted a hypothesis when it was 10000 times more likely than the other hypothesis (Bayes Factor threshold T = 10000). Our experiments with Wald’s SPRT used Type I and II error bounds of 0.01. We chose an indiﬀerence region δ so as to make the Type I and Type II errors for both the Wald’s Test and the Bayes Factor test equal. The statistical estimation engine of the PRISM model checker always needed 92042 samples to estimate the probability of the BLTL formulae being true. The results of experiments with the Fibroblast Growth Factor Signaling Model (see [24], [25] for details) are presented. We checked the property whether the probability that Grb2 binds to FRS2 within 20 time units exceeds θ (for several values of θ): H0 : M = P≥θ [ F20 (F RS2 GRB > 0 )] The power curves and the number of samples for this benchmark are plotted in Fig. 2(a) and Fig. 2(b) respectively. A power curve indicates the probability of accepting the null hypothesis for various values of the threshold probability θ in the PBLTL formula. We chose the Wald’s Test so that its power curve matched that of the Bayesian Test at the 0.01 and 0.99 acceptance probability. The goal is to make sure that the two tests have equal statistical power. From Figure 2(b), it is clear that both the power curves are almost on top of each other and hence, both the tests have indeed been calibrated to be equally powerful. The Bayesian algorithm needs fewer samples than Wald’s SPRT test for this benchmark. This shows that the Bayesian Statistical Model Checking performs better than an approach based on Wald’s SPRT. We also studied the continuous time Markov Chain model [5,41] for circadian rhythm. We checked the property that the probability of the number of activated messenger RNAs exceeding 5 units within 0.25 time units is more than θ (for various values of θ): H0 : M = P≥θ [ F0.25 (ma > 5) ]
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500 Bayesian Test Wald’s Test
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1
350
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Probability of Acceptance
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400
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Bayesian Test Wald’s Test
0.7 0.6 0.5 0.4 0.3 0.2
50
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0
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0
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1
(a) Number of Samples for various probability (b) Power Curve of the Bayesian and thresholds in the formula Wald’s approach Fig. 1. Fibroblast Growth Factor Signaling Model: The system satisﬁes the formula with probability 0.58. (Bayes Factor=10000)
700 Bayesian Test Wald’s Test 1 Bayesian Test
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Number of Sampled Needed
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(a) Number of Samples for various probability (b) Power Curve of the Bayesian and thresholds in the formula Wald’s approach Fig. 2. Circadian Rhythm: The system satisﬁes the formula with probability 0.93. (Bayes Factor=10000)
The power curves and the number of samples for this benchmark are plotted in Fig. 2(b) and Fig. 2(a) respectively. We calibrated Wald’s test so that its power curve closely matched that of the Bayesian Test so as to make a fair comparison. From the ﬁgure, we observe that the Bayesian algorithm always needs fewer samples than the Wald’s SPRT test for this benchmark. We also analyzed the model on Cell cycle control [33] and studied the probability that Cyclin gets bound within the ﬁrst 0.5 time units. We check the property that the probability of the number of bound Cyclin molecules exceeds 3 units within 0.5 time units exceeds θ (for various values of θ):
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800 Bayesian Test Wald’s Test
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1 Bayesian Test
0.9
Wald’s Test
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500 Probability of Acceptance
Number of Samples
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(a) Number of Samples for various probability (b) Power Curve of the Bayesian and thresholds in the formula Wald’s approach Fig. 3. Cell Cycle Control: The system satisﬁes the formula with probability 0.34. (Bayes Factor=10000)
H0 : M = P≥θ [ F0.5 (cyclin bound > 3) ] The results of our experiment are presented in Fig. 3(a). The Bayesian Statistical Model Checking algorithm usually required fewer samples than the approach based on Wald’s SPRT. 4.2
SBML Experiments
We also studied SBML models using the implementation of Gillespie’s Stochastic Simulation Algorithm in Matlab’s Systems Biology Toolbox. We analyzed two large models with over 108 and 1017 species. We used monitors written in Matlab to verify the BLTL properties on traces. Our analysis of the experiments in this section is purely Bayesian, i.e., we have studied the performance of the algorithm over only one run (using uniform priors). In the previous sections, we had compared the performance of our algorithm with Wald’s SPRT by running the algorithm several times on the same model  a frequentist approach. We analyzed the Yeast Heterotrimeric G Protein Cycle benchmark [44]. We analyzed the property that the G protein stays above the threshold of 6000 units for 2 time units and falls below 6000 before 20 time units. H0 : M = P≥θ [ G2 (GP rotein > 6000) and F20 (GP rotein < 6000)] . We also ran experiments using the Lotka model [22] and veriﬁed the property that the number of copies of the x species rises to a threshold level within 0.01 time units. H0 : M = P≥θ [ F0.01 (x > 1.4 ∗ 107 )]
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S.K. Jha et al. Table 1. Performance on the G Protein (left) and Lotka Benchmark (right)
Probability # Samples Needed 0.2 3 0.6 8 0.8 14 0.9 23 0.9999 99
Probability # Samples Needed 0.1 2 0.5 6 0.7 10 0.9 23 0.99 69
3.5
6000
Non−informative Prior Informative Prior Misleading Prior
3
Non−informative Prior Informative Prior Misleading Prior
5000
Number of Samples
Probability Density
2.5
2
1.5
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0
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1
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1
(a) Shape of the Priors used in our Exper (b) Number of Samples with Diﬀerent iments Classes of Priors
Probability of Accepting the Null Hypothesis
1 Non−informative Prior Informative Prior Misleading Prior
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
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1
(c) Power curve of the tests of the Algorithm Fig. 4. Diﬀerent Classes of Priors
The results of our experiments are shown in Table 1: both hypotheses are always accepted, although the number of samples increases with the probability threshold of the temporal formula.
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231
Experiment with Diﬀerent Classes of Priors
We investigated the eﬀect of priors on the performance of the Bayesian Model Checking algorithm. We used three diﬀerent priors  noninformative prior, an informative prior and a misleading prior. The priors, the number of samples needed by the Bayesian algorithm for these priors, and the power curve for each of these priors is also plotted in Fig. 4(a), Fig. 4(b) and Fig. 4(c) respectively. The priors used are Beta distributions with diﬀerent shape parameters: (i) α = 1/2, β = 1/2: noninformative prior, (ii) α = 1.4, β = 2 : informative prior with a peak around 0.34 (iii) α = 2, β = 2: a misleading prior with peak around 0.5. Fig. 4(b) shows that the number of samples needed by the Bayesian algorithm becomes smaller when the prior probability distribution is informative and supports the true hypothesis. Also, the power curve (see Fig. 4(c)) becomes sharper when the Bayesian algorithm is given a correct and informative prior probability distribution. A completely noninformative prior also performs well both in the number of samples and the power of the test. Strongly misleading priors make the power curve less steep. However, the algorithm still performs quite well when the actual probability of the system is away from the threshold probability in the formula.
5
Conclusions and Future Work
We have introduced the ﬁrst algorithm for Probabilistic Model Checking based on Bayesian Sequential Hypothesis Testing. Our algorithm terminates with probability 1, and provides bounds on the probability of returning an incorrect answer. Empirically, we have shown that our algorithm requires fewer traces to terminate than techniques based on Classical Statistics. This is not surprising as the Bayesian method comparing composite hypotheses whereas techniques like Wald’s SPRT are comparing simple hypotheses. This advantage in eﬃciency is important in the context of Systems Biology as the cost of generating traces is not necessarily negligible. Bayesian methods also aﬀord a convenient means for incorporating domain knowledge through the prior distributions. Our algorithm is presently limited to incorporating prior information on the probability that the property is true. A more fully Bayesian approach would incorporate prior information on not just the property, but also the starting state and parameters of the model. We are presently extending our method to address this limitation.
Acknowledgments The authors would like to thank H.L.S. Younes for comments on a draft of this paper.
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18. Fages, F.: Temporal logic constraints in the biochemical abstract machine biocham. In: Hill, P.M. (ed.) LOPSTR 2005. LNCS, vol. 3901, pp. 1–5. Springer, Heidelberg (2006) 19. Finkbeiner, B., Sipma, H.: Checking ﬁnite traces using alternating automata. In: Proceedings of Runtime Veriﬁcation (RV 2001), pp. 44–60 (2001) 20. Fisher, J., Piterman, N., Hubbard, E.J., Stern, M.J., Harel, D.: Computational insights into caenorhabditis elegans vulval development. Proc. Natl. Acad. Sci. U S A 102(6), 1951–1956 (2005) 21. Ghosh, B., Sen, P. (eds.): Handbook of sequential analysis. Dekker, New York (1991) 22. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry 81(25), 2340–2361 (1977) 23. Grosu, R., Smolka, S.: Monte Carlo Model Checking. In: CAV, pp. 271–286 (2005) 24. Heath, J., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic model checking of complex biological pathways. In: Priami, C. (ed.) CMSB 2006. LNCS (LNBI), vol. 4210, pp. 32–47. Springer, Heidelberg (2006) 25. Heath, J., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic model checking of complex biological pathways. Theoretical Computer Science 319(3), 239–257 (2008) 26. H´erault, T., Lassaigne, R., Magniette, F., Peyronnet, S.: Approximate probabilistic model checking. In: Steﬀen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 73–84. Springer, Heidelberg (2004) 27. Hinton, A., Kwiatkowska, M., Norman, G., Parker, D.: PRISM: A tool for automatic veriﬁcation of probabilistic systems. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 441–444. Springer, Heidelberg (2006) 28. Jeﬀreys, H.: Theory of Probability. Clarendon Press, Oxford (1961) 29. Jha, S.K., Clarke, E.M., Langmead, C.J., Legay, A., Platzer, A., Zuliani, P.: A bayesian approach to model checking biological systems. Technical Report CMUCS09110, Computer Science Department, Carnegie Mellon University (2009) 30. Kam, N., Harel, D., Cohen, I.R.: Modeling biological reactivity: Statecharts vs. boolean logic. In: Proceedings of the Second International Conference on Systems Biology (2001) 31. Kwiatkowska, M.Z., Norman, G., Parker, D.: Prism 2.0: A tool for probabilistic model checking. In: QEST, pp. 322–323. IEEE, Los Alamitos (2004) 32. Langmead, C.J.: Generalized Queries and Bayesian Statistical Model Checking in Dynamic Bayesian Networks: Application to Personalized Medicine. In: Proc. 8th Ann. Intnl Conf. on Comput. Sys. Bioinf. (CSB), pp. 201–212 (2009) 33. Lecca, P., Priami, C.: Cell cycle control in eukaryotes: A BioSpi model. In: Proc. Workshop on Concurrent Models in Molecular Biology (BioConcur 2003). ENTCS (2003) 34. McAdams, H., Shapiro, L.: Circuit simulation of genetic networks. Science 269, 650–656 (1995) 35. Owicki, S.S., Lamport, L.: Proving liveness properties of concurrent programs. ACM Trans. Program. Lang. Syst. 4(3), 455–495 (1982) 36. Pnueli, A.: The temporal logic of programs. In: FOCS, pp. 46–57. IEEE, Los Alamitos (1977) 37. Priami, C., Regev, A., Shapiro, E., Silverman, W.: Application of a stochastic namepassing calculus to representation and simulation of molecular processes. Inf. Process. Lett. 80(1), 25–31 (2001)
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Dynamic Compartments in the Imperative πCalculus Mathias John1 , C´edric Lhoussaine2,4 , and Joachim Niehren3,4 1
University of Rostock, Computer Science, Modeling and Simulation Group 2 University of Lille 1 3 INRIA, Lille, Mostrare 4 BioComputing, LIFL (CNRS UMR8022) and IRI (CNRS USR3078)
Abstract. Dynamic compartments with mutable conﬁgurations and variable volumes are of basic interest for the stochastic modeling of biochemistry in cells. We propose a new language to express dynamic compartments that we call the imperative πcalculus. It is obtained from the attributed πcalculus by adding imperative assignment operations to a global store. Previous approaches to dynamic compartments are improved in ﬂexibility or eﬃciency. This is illustrated by an appropriate model of osmosis and a correct encoding of BioAmbients.
1
Introduction
Concurrent control is crucial for the stochastic modeling of biochemical processes in living cells [19,2,13]. The regulation of such systems depends on all kinds of physical or chemical aspects, such as volume, surface, temperature, pressure, pH value, spatial coordinates and structures. Most of these aspects are of global nature, so they require modeling languages in which global concurrent control can be expressed [20]. In this paper, we present a new modeling language, that permits to express many aspects with global control in a uniform manner, and illustrate its usefulness by modeling dynamic compartments with mutable volumes and surfaces. Dynamic compartments may change their nesting structure dynamically, by applying operations for compartment creation, removal and merging. These operations may inﬂuence the speed of diverse reactions within compartments, in particular when compartment volumes change (global to local interactions). Vice versa, local reactions within a single compartment may eﬀect global numeric attributes such as volume and surface (local to global interaction). Various languages for modeling systems with dynamic compartments were proposed for systems biology [18,14,21], but none of them can express physical, chemical, and compartimental aspects in a uniform manner, while providing eﬃcient stochastic simulation. Spatial languages such as the Brane Calculi [2] or BioAmbients [18] ﬁx a particular set of operators on compartments, and provide a special purpose solution for these operations. The πcalculus with polyadic synchronization and global priorities π@ is more ﬂexible, in that it permits to P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 235–250, 2009. c SpringerVerlag Berlin Heidelberg 2009
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encode all kinds of compartment structures, including those of Brane Calculi and BioAmbients [20]. Unfortunately, such prioritybased encodings are complex, low level, and ineﬃcient. Consider e.g. the dissolving of a compartment with n equal molecules. Informing all of them requires O(n) interactions rather than O(1) by updating all at once. Furthermore, π@ lacks general support for stochastic rates and numeric attributes such as volumes and pHvalues. The only solution to compartments with variable volumes so far [21] was expressed in the special purpose dialect called Sπ@. Numerical attributes of compartments are equally lacking in Bigraphs [14,12], a modeling language for spatial dynamics based on a particular form of hypergraph rewriting. Thus, the question is whether there exists a better general purpose language for expressing dynamic compartments. In this paper, we start from the attributed πcalculus [11], and enrich it by an imperative store for global control. The attributed πcalculus is parametrized by a sequential higherorder language L for describing all kinds of values (symbolic and numeric) and constraints. It features “attributed” processes A(e1 , . . . , en ) with values deﬁned by expressions e1 , . . . , en of L. For instance, cells with variable volumes vol can be modeled by using a single attribute: Cell(vol) enter[λr. if r 0 and the time points of interest is ˆ where dˆ · Δt is assumed to be the set {d · Δt} with d ranging over {0, 1, . . . , d} the maximal time point of interest. Next we assume that the values of the variables can be observed with only ﬁnite precision and accordingly partition the range of each xi into Li intervals [vimin , vi1 ), [vi1 , vi2 ), . . . , [viLi −1 , vimax ]. We denote this set of intervals as Ii . We also
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similarly discretize the range of each parameter pj into a set of intervals denoted as In+j . The set I = {Ii }1≤i≤n ∪ {In+j }1≤j≤m is called the discretization. As pointed out earlier, the initial values vector as well as the rate constants (even when they are known) will be given not as point values but as distributions (usually uniform) over the intervals deﬁned by the discretization. We correspondingly assume we are given a prior distribution in the form of a probability density function Υ 0 capturing the distribution of initial values. For example, suppose we are given that the initial values are uniformly distributed within a hypercube Iˆ1 × Iˆ2 × . . . × Iˆn+m , where Iˆi ∈ Ii for each i. Let Iˆi = [li , ui ) and w ˆi = u i − li . Then the corresponding prior probability density function Υ 0 will be given by: 1 if z ∈ Iˆ1 × Iˆ2 × . . . × Iˆn+m , 0 Υ (z) = wˆ1 ·wˆ2 ·...·wˆn+m 0 otherwise. The associated probability space we have in mind is (D, BD , P 0 ) where BD is the Borel σalgebra over D; the minimal σalgebra containing the open sets of D under the usual topology. P 0 is the probability distribution induced by Υ 0 and is given by: P 0 (B) = B Υ 0 (z)dz, for every B ∈ BD . Further, T RAJideal = {Z(t)}t≥0 with Z(0) ∈ Iˆ1 × Iˆ2 × . . . × Iˆn+m is the family of trajectories starting from all the possible points in this hypercube. Since the ﬂow is continuous and hence measurable we can associate a probability distribution P t over BD for every t. To deﬁne this, let Φ−1 t (B) = {z  Φ(t, z ) ∈ B} for −1 B ∈ BD . Since Φ(t, ·) is measurable, we have Φt (B) ∈ BD too. We can now deﬁne P t as: P t (B) = P 0 (Φ−1 t (B)), for every B ∈ BD . Let v be in the range of xi . We deﬁne [v] as the interval in which v falls. In other words, [v] = I iﬀ v ∈ I. Similarly, [k] = J if k ∈ J for a parameter value k of pj with J ∈ In+j . Lifting this notation to the vector setting, if z = (v1 , v2 , . . . , vn , k1 , k2 , . . . , km ) ∈ IRn+m , we deﬁne [z] = ([v1 ], [v2 ], . . . , [vn ], [k1 ], . . . , [km ]) and refer to it as a + discrete state. An MCstate is a pair (s, d), where s is a discrete state and ˆ d ∈ {1, 2, . . . , d}. We next deﬁne P r(s, d) = P d·Δt ({z  z ∈ I1 × I2 × . . . × In+m }), where s = (I1 , I2 , . . . , In+m ). We term the MCstate M to be feasible iﬀ P r(M ) > 0. The transition relation denoted as →, between MCstates is deﬁned via: M = (s, d) → M = (s , d ) iﬀ d = d + 1 and both M and M are feasible and there exist z0 , z, and z such that Φ(d · Δt, z0 ) = z and Φ((d + 1) · Δt, z0 ) = z . Furthermore, [z] = s and [z ] = s . Let E, F denote, respectively, the event that the system is in the discrete state s at time d · Δt and in the discrete state s at time (d + 1) · Δt for two feasible MCstates (s, d · Δt) and (s , (d + 1) · Δt). Let EF = E ∩ F denote joint event {z0  Φ(d · Δt, z0 ) ∈ s, Φ((d + 1) · Δt, z0 ) ∈ s }. Consequently, we deﬁne the
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transition probability P r((s, d) → (s , d )) = P r(F E) = P r(EF )/P r(E). Since P r(E) > 0 this transition probability is welldeﬁned. Let M = {M1 , M2 , . . . , Mnˆ } be the set of Mstates. We can now deﬁne the Markov chain MC ideal = (M, {pij }) with transition probabilities pij = P r(Mi → Mj ) as above. 2.3
The Markov Chain MC approx
MC ideal can not be explicitly computed. Hence we sample z0 a suﬃciently larger number of times, say N , according to the prior distribution P 0 (we say more about N below). For each sampled initial z0 , we determine through numerical ˆ We also integration the Mstates [Φ(d·Δt, z0 )], with d ranging over {0, 1, . . . , d}. determine the transitions along this trajectory. Then through a simple counting process involving these N trajectories, we compute a Markov chain that we refer to as the MC approx . Since N is ﬁnite, there will be an error between the transition probabilities (also the MCstate probabilities) computed using MC approx and the ones deﬁned by MC ideal . By the central limit theorem [18], this error can be probabilistically bounded. In other words, given an error bound and a conﬁdence level c, we can compute N , the number of samples required to get an error less than or equal to with likelihood c (the Appendix gives more details). Further, this error will tend to 0 with probability 1 as N tends to ∞. There will be an additional error induced by the pthorder numerical integration method we use to compute the N trajectories. This error will tend to 0 as Δt tends to 0 or p tends to ∞. However, the number of states of this Markov chain will be exponential in n and hence for many signaling pathways MC approx will be too large a structure. Hence we shall construct a timevariant DBN called the BDM to compactly represent MC approx . We shall however compute the BDM directly from the N sampled trajectories. 2.4
The BDM Representation
In what follows, we assume the basic background concerning Bayesian networks and dynamic Bayesian networks [3]. The graphical structure of the DBN used for our approximation can be derived from the diﬀerential equations. It will have n+ m random variables (corresponding to the variables and the parameters) as nodes ˆ For convenience, we for each time slice d · Δt with d ranging over {0, 1, . . . , d}. will use the same name to denote a variable (parameter) and the corresponding random variable. From the context it should be clear which role is intended. The random variable xi (pj ) can assume as values, the ﬁnite set of intervals Ii (In+j ). The variable (parameter) xi (pj ) in the time slice d · Δt will be written as xdi d (pj ). Edges connecting a node in the dth slice to a node in the (d+1)th slice will be determined by the dependencies of the variables and the parameters in the ODEs. Suppose zld is a (variable or parameter) node in the dth time slice and zqd+1 is a node in the next time slice. Then there will be an edge from zld to zqd+1
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k3
S + E ES − −→ E + P k2
dS dt dE dt dES dt dP dt
= −k1 · S · E + k · ES = −k1 · S · E + (k2 + k3 ) · ES = k1 · S · E − (k2 + k3 ) · ES = k3 · ES
Fig. 1. The ODE model of the enzymekinetic system and its BDM
iﬀ zl = zq or zq is a variable node and zl appears in the expression for z˙q in the system of ODEs. As usual, the parents of the node zqd+1 will be the set of nodes of the form zld from which there is an edge into zqd+1 . Suppose, parents(xd+1 )= i {z1d, . . . , zld }. Then conditional probability table (CPT) associated with the node xd+1 will have entries of the form P r(xd+1 = I  z1d = I 1 , . . . , zld = I l ) = h with i i k I ranging over Ii and I ranging over Ik and h ∈ [0, 1]. For instance, Figure 1 shows two adjacent slices of a enzymekinetic system. In this BDM, the parent nodes of P d+1 are P d , ES d and k3d . As mentioned earlier, the parameters are assumed to not change their values during a run and hence we denote kid as simply ki and there will be no CPTs associated with these nodes. As illustrated by the example, the connectivity between the nodes in successive slices will remain invariant. However, due to the fact that the CPTs associated with the nodes capture the transition probabilities of MCstates, they will be time variant. ˆ n ) states and O((dˆ − 1)K 2n ) MC approx will have, in the worst case, O(dK transitions, where K is the maximum of Ii  with 1 ≤ i ≤ n + m. In contrast, the ˆ + m)) and the conditional number of nodes in the BDM representation is O(d(n probability table associated each node will have at most O(K R+1 ) entries, where R is the maximal number of parents a node can have. Usually, the reactants in pathway models will be sparsely coupled to each other and hence R will be much smaller than n. For instance, in the case study to be presented, n = 32 and R = 5. Even in cases where R is large, due to the nature of the ODEs we deal with, we can often break up the corresponding node into nodes with smaller fanin degrees and thus reduce R [16]. To ﬁll up the entries of the CPTs associated with the nodes we randomly choose N combinations of initial values for the variables and the parameters from their prior distribution as before. (If we want a coverage of J samples per interval in an n + m dimensional vector of intervals, then by exploiting the network structure, we can make do with N = JK R+1 samples.) We then perform numerical integration to generate N trajectories and discretize those trajectories by the predeﬁned intervals and compute the conditional probabilities for each node by simple counting. For example, suppose α trajectories hit (P d =
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I , ES d = I , k3 = I ) and β of them in turn hit (P d+1 = I), then P r(P d+1 = β IP d = I , ES d = I , k3 = I ) = α . Further, the MC approx can be easily recovered from this DBN [19]. In this sense, our BDM representation is a principled probabilistic approximation of the dynamics induced by the system of ODEs. Various optimizations can be developed to reduce the practical complexity of the BDM construction. The details can be found in [16]. Though the construction of the BDM involves signiﬁcant computational eﬀort, it is a one time cost. Moreover, a substantial portion of the computation can be executed in parallel. Further, once the BDM has been constructed, many of the analysis tasks can be performed very eﬃciently and the one time cost of constructing the BDM can be easily amortized. We present some experimental results in support of this in the next section.
3
Analysis
We now present some of the analysis techniques that we have developed so far for the BDM representation. These techniques are based on the basic Bayesian inference method called the F F (Factored Frontier) algorithm [15] and can be used to answer elementary probabilistic queries as well for performing parameter (rate constants) estimation and sensitivity analysis. Our goal here is not to develop new algorithms to solve these problems. Rather, we wish to demonstrate how standard techniques for tackling these problems can be adapted to BDM framework in a straightforward manner. We validate our techniques using a relatively large signaling pathway and show the relevant experimental results along with our techniques. 3.1
The EGFNGF Signaling Pathway and Its BDM
PC12 cells are a valuable model system in neuroscience. They proliferate in response to EGF stimulation but diﬀerentiate into sympathetic neurons in response to NGF. This interesting phenomenon has been intensively studied [20]. It has been reported that the signal speciﬁcity is correlated with diﬀerent Erk dynamics. Speciﬁcally, a transient activation of Erk1/2 has been associated with cell proliferation, while a sustained activity has been linked to diﬀerentiation. How EGF and NGF aﬀect the dynamics of active Erk through a network of intermediate signaling proteins is shown schematically in Figure 2. This model not only includes a common pathway to Erk through Ras shared by both the EGFR and NGFR, but also includes two important side branches through PI3K and C3G, which introduce multiple feedback loops thus complicating the dynamics. The ODE model of this pathway is available in the BioModels database1 . It consists of 32 diﬀerential equations and 48 associated rate parameters (estimated from multiple sets of experimental data). 1
http://www.ebi.ac.uk/biomodels/
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... ... d[f reeEGF R ] = 0 .0121008× [boundEGF R ] dt 0 .0000218503×[ EGF ] × [ f reeEGF R ] d[boundEGF R ] = 0 .0000218503× [ EGF ] × [ f reeEGF R ] dt 0.0121008 × [boundEGF R ] d[Sos] 1611.97× [P90Rsk* ] × [Sos] = dt [Sos] + 896896 694.731× [boundEGF R ] × [Sos] [Sos] + 6086070 389.428 × [boundN GF R ] × [Sos] [Sos] + 211.266 d[Sos*] 694.731× [boundEGF R ] × [Sos] = dt [Sos] + 6086070 389.428× [boundN GF R ] × [Sos] + [Sos] + 211.266 1611.97× [P90Rsk* ] × [Sos] [Sos] + 896896 ... ...
Fig. 2. EGFNGF pathway [1]
To construct the BDM, we ﬁrst derived its graph from its ODEs. We then discretized the ranges of each variable and parameter into 5 equalsize intervals and ﬁxed the time step Δt to be 1 minute. Our experimental data (western blot) is such that 5 uniform intervals seems an appropriate choice. However our construction can be easily extended to nonuniform values intervals and time points. To ﬁll up the conditional probability tables associated with the nodes, 3 × 106 trajectories were generated up to 100 mins by sampling initial states and parameters from the prior which are assumed to be uniform distributions over certain intervals (see [16]). The computational workload was distributed on 10 Opteron 2.2GHz processors in a cluster. It took around 4 hours to construct the BDM. All the subsequent experiments reported below were done using an Intel Xeon 2.8GHz processor. 3.2
Probabilistic Inference
As pointed out earlier, although the dynamics deﬁned by the ODEs is deterministic, to answer a basic query such as “what will be the concentration of the protein xi at time t? ” one will have to numerically generate a representative sample of trajectories and compute the average of the values for xi at t yielded by the individual trajectories. Using our BDM approximation, we can answer such a basic query and other more sophisticated queries by Bayesian inference. Speciﬁcally, given a Bayesian network, some observed evidence and some knowledge about the distribution of values of a set of variables, Bayesian inference aims to compute posterior distribution for a set of query variables. In our setting, the observed evidence refers to the initial conditions, known parameters, and experimental data. Query variables potentially refer to all the random variables in the BDM. We adopt the approximate algorithm known as the Factored Frontier (FF) algorithm [15]. It approximates joint distributions over each time point as a product of marginal
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distributions and computes the posterior distribution according to: P r(xdi D) = (P r(xdi P a(xdi ) = I) P r(uD)). I
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Here P r(uD) are the marginal distributions over the parents, D is the evidence, and P a(xi ) denotes the parents of xi . The implementation of FF is straightforward. By storing P r(xdi P a(xdi )) in the conditional probability tables and propagating P r(uD) to the next time point, we can use equation 1 to compute ˆ + m)K R+1 ), where K P r(xdi D). The time complexity of this algorithm is O(d(n is the maximal number of intervals associated with a variable or rate constant’s value domain. Further, R is the maximal number of parents a node can have. Using this algorithm, and with some additional simple computations, many queries can be answered. For instance, we identify each interval I = [l, u) in I with its midpoint l+u 2 . Then after inferring the probability distribution of xi over intervals, the expected value E(xdi ) at a time slice d can be computed and used to validate the model by comparing it with the cell population based data that may be available for xi at d · Δt. To test the quality of our approximation, we implemented Monte Carlo integration for the ODE model to get good estimates by sampling. Speciﬁcally, we numerically generated a number of random trajectories according to the priorusing ODEs, discretized them and computed the average values of the variables at the chosen time points. Our experiments show that the average values converge when the number of random trajectories generated is roughly 104 . The averaged trajectories projected to individual protein concentration time series values are termed to be the nominal simulation proﬁles. Using the implemented FF algorithm, the mean of each variable over time was computed. The resulting time proﬁles are termed to be the BDMsimulation proﬁles. As summarized in Figure 3, our BDMsimulation proﬁles ﬁt the nominal simulation proﬁles quite well for most of the cases. In terms of running time, a single execution of FF inference requires 0.29 seconds while generating a stable nominal proﬁle requires 386.4 seconds. Thus,
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the total computation time will be sharply reduced by our approach when many such “queries” need to be answered. In the next subsection, we will further demonstrate this advantage by carrying out a simulationintensive analysis task. 3.3
Parameter Estimation
Lack of knowledge about the parameters and hence the need to perform parameter estimation using limited data has long been identiﬁed as a major bottleneck of pathway modeling. Current approaches to parameter estimation formulate it as a nonlinear optimization problem [21]. A typical procedure will involve searching in a high dimensional solution space, in which each point represents a vector of parameter values. Whether a point is good or not is measured by the objective function, which will capture the diﬀerence between experimental data and prediction generated by simulations using the corresponding parameters. For a large pathway model, one often needs to evaluate a very large number of solution points involving a numerical integration for each evaluation. This makes the whole process computationally intensive. The BDM representation allows us to carry out the search for good parameter values in a hierarchical manner. Due to the discretized nature of the BDM, the solution space is transformed to a rectilinear grid consisting of a spaceﬁlling tessellation by hyperrectangles that we call blocks. An important observation is that kinetic parameters are often robust [22]. In other words, the points around the best solution in the search space will also have relatively small objective values. Thus, instead of searching point by point in the solution space, we can ﬁrst search for a few promising blocks and then take a closer look within these small blocks. Therefore, the general scheme of our “grid search” algorithm will consist of two phases: (1) identify good blocks, (2) do local search within candidate blocks. We note that phase(2) is necessary only when we aim to estimate parameters with ﬁner granularity than the granularity of the BDM’s discretization. Otherwise, one can skip phase(2) and return a probabilistic estimate (typically a Maximal Likelihood Estimate) of a combination of intervals of parameter values. For executing phase(1), we can apply any standard search algorithms over the discretized search space. As the discretized search space is much smaller than the original one, simple direct search algorithm such as Hooke & Jeeves’s search [23] can be adopted and the overall search process will only require a small number of executions of the FF algorithm. In order to test the performance of the BDMbased parameter estimation method, we synthesized experimental time series data for 9 (out of 32) proteins {bounded EGFR, bounded NGFR, active Sos, active C3G, active Akt, active p90RSK, active Erk, active Mek, active PI3K}, measured at the time points {2, 5, 10, 20, 30, 40, 50, 60, 80, 100} (min).This data was synthesized using prior knowledge about initial conditions and parameters [16]. To mimic western blot data which is cell population based, we ﬁrst averaged 104 random trajectories generated by sampling initial states and rate constants, and then added observation noise with variance 5% to the simulated values. With the assumed measurement precision, those values were discretized into 5 intervals, which represent the concentration levels in western blot data. We reserved the data of 7
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proteins for training the parameters and reserved the rest data for testing the quality of the estimated parameter values. Assuming that 20 of the 48 parameter values are unknown, a modiﬁed version of Hooke & Jeeves algorithm was implemented to search for in the discretized parameter space. The parameters obtained can be found in [16]. As shown in Figure 4, the BDMsimulation proﬁles generated using the estimated parameters obtained (with the match to training data as shown) has good agreement with the test data. We compared the eﬃciency and quality of our results with the following ODEs based optimization algorithms: LevenbergMarquardt (LM), Genetic Algorithm (GA), Stochastic Ranking Evolutionary Strategy (SRES), and Particle Swarm Optimization (PSO). These optimization algorithms were executed using the COPASI [24] tool. We scored the resulting parameters obtained from all the algorithms using the weighted sumofsquares diﬀerence between the experimental data and the corresponding simulation proﬁles (i.e. low scores correspond to low errors). The results are summarized in Figure 5, which suggests that our method achieves a good balance between accuracy and performance. We also note that the cost of constructing the BDM representation gets rapidly amortized. In fact
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the savings become even more signiﬁcant when we perform additional analysis tasks such as sensitivity analysis. 3.4
Global Sensitivity Analysis
Sensitivity analysis has been used to identify the critical parameters in signal transduction. To overcome the limitations of traditional local sensitivity analysis methods global methods have been proposed recently, e.g. multiparametric sensitivity analysis (MPSA) [25]. The MPSA procedure consists of: (1) draw samples from parameter space and for each combination of parameters, compute the weighted sum of squared error between experimental data and predictions generated by selected parameters; (2) classify the sampled parameter sets into two classes (good and bad) using a threshold error value; (3) plot the cumulative frequency of the parameter values associated with the two classes; (4) evaluate the sensitivities as the KolmogorovSmirnov statistic of cumulative frequency curves. To improve this process, [25] adopts Latin hypercube sampling (LHS) since it requires fewer samples while guaranteeing that individual parameter ranges are evenly covered. In our BDM setting, MPSA can be done in a similar manner using LHS since the parameter space is discretized into blocks. In addition, the number of samples used to reach convergence is reduced since we can quickly evaluate the whole block instead of having to draw samples from a block. We modiﬁed and implemented the MPSA method for the BDM. Using the same experimental data set introduced in previous subsection, the global sensitivities (KS statistics) of the parameters were computed. The results are shown in Figure 6. The cumulative frequency distributions for the acceptable and unacceptable cases of the rate constants can be found in [16]. Speciﬁcally, the reactions involved in the phosphorylation of Erk (k23 ), Mek (k17 ), Akt (k34 ) and p90RSK (k28 ) have the highest sensitivities, indicating that these reactions aﬀect the system behavior most directly. These results are consistent with previous ﬁndings [20]. The MPSA method adopts Monte Carlo strategy for the ODE model. We recorded the running time of the algorithm till the KS values converged. The total running time of the ODEs based MPSA method was about 22 hours, while the MPSA method based on the BDM required only 34 minutes.
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Discussion
We have proposed a probabilistic approximation scheme for signaling pathway dynamics speciﬁed as a system of ODEs. Given a discretization and an initial distribution, it consists of precomputing and storing a representative sample of trajectories induced by the system of ODEs. We use a dynamic Bayesian network representation, called the Bayesian Dynamics Model, to compactly represent these trajectories by exploiting the pathway structure. Basically, the underlying graph of the BDM captures the dependencies of the variables on other variables and rate constants as deﬁned by the system of ODEs. Due to the probabilistic graphical representation, a variety of analysis questions concerning the pathway dynamics traditionally addressed using Monte Carlo simulations can be converted to Bayesian inference and solved more eﬃciently. Using the FF algorithm for doing basic Bayesian inference, we have adapted standard parameter estimation and sensitivity analysis algorithms to the BDM setting. We have demonstrated the applicability of our techniques with the help of the good sized EGFNGF signaling pathway. A number of further lines of work suggest themselves. Firstly, we need to apply our method to a variety of pathway models. We are currently doing so in collaboration with biologists. Secondly, it will be useful to augment the ODE model with some discrete features but this should be easy to achieve. A more challenging issue is to abstract the BDM representation to an inputoutput transducer so that one can eﬃciently model networks of pathways and intercellular interactions models. Finally, it will be important to develop formal veriﬁcation techniques based on the BDM representation. In this context, it is worth noting that the FF algorithm can compute the marginal probabilities of the discretized values of variables at speciﬁc time points. Hence a good starting point will be to develop probabilistic bounded model checking methods for speciﬁcations based on the BDM model.
References 1. Brown, K.S., Hill, C.C., Calero, G.A., Lee, K.H., Sethna, J.P., Cerione, R.A.: The statistical mechanics of complex signaling networks: nerve growth factor signaling. Phys. Biol. 1, 184–195 (2004) 2. Matsuno, H., Tanaka, Y., Aoshima, H., Doi, A., Matsui, M., Miyano, S.: Biopathways representation and simulation on hybrid functional Petri net. Silico Biol. 3(3), 389–404 (2003) 3. Murphy, K.P.: Dynamic Bayesian Networks: Representation, Inference and Learning. PhD thesis, University of California, Berkeley (2002) 4. Antoniotti, M., Policriti, A., Ugel, N., Mishra, B.: XSsystems: extended ssystems and algebraic diﬀerential automata for modeling cellular behavior. In: Sahni, S.K., Prasanna, V.K., Shukla, U. (eds.) HiPC 2002. LNCS, vol. 2552, pp. 431–442. Springer, Heidelberg (2002) 5. de Jong, H., Page, M.: Search for steady states of piecewiselinear diﬀerential equation models of genetic regulatory networks. IEEE/ACM T. Comput. Bi. 5(2), 208– 223 (2008)
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6. Ghosh, R., Tomlin, C.: Symbolic reachable set computation of piecewise aﬃne hybrid automata and its application to biological modelling: Deltanotch protein signalling. Systems Biol. 1(1), 170–183 (2004) 7. Calder, M., Gilmore, S., Hillston, J.: Modelling the inﬂuence of RKIP on the ERK signalling pathway using the stochastic process algebra PEPA. In: Priami, C., Ing´ olfsd´ ottir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on Computational Systems Biology VII. LNCS (LNBI), vol. 4230, pp. 1–23. Springer, Heidelberg (2006) 8. Calder, M., Vyshemirsky, V., Gilbert, D., Orton, R.: Analysis of signalling pathways using continuous time Markov chains. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 44–67. Springer, Heidelberg (2006) 9. Ciocchetta, F., Degasperi, A., Hillston, J., Calder, M.: CTMC with levels models for biochemical systems. Elsevier, Amsterdam (2009) (preprint submitted) 10. Nodelman, U., Shelton, C.R., Koller, D.: Continuous time Bayesian networks. In: Proceedings of the 18th Conference in Uncertainty in Artiﬁcial Intelligence, Alberta, Canada, pp. 378–387 (2002) 11. Langmead, C., Jha, S., Clarke, E.: Temporal logics as query languages for dynamic Bayesian networks: Application to D. Melanogaster embryo development. Technical report, Carnegie Mellon University (2006) 12. Clarke, E.M., Faeder, J.R., Langmead, C.J., Harris, L.A., Jha, S.K., Legay, A.: Statistical model checking in BioLab: Applications to the automated analysis of TCell receptor signaling pathway. In: Heiner, M., Uhrmacher, A.M. (eds.) CMSB 2008. LNCS (LNBI), vol. 5307, pp. 231–250. Springer, Heidelberg (2008) 13. Heath, J., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic model checking of complex biological pathways. Theor. Comput. Sc. 319(3), 239–257 (2008) 14. Geisweiller, N., Hillston, J., Stenico, M.: Relating continuous and discrete PEPA models of signalling pathways. Theor. Comput. Sc. 404(2), 97–111 (2008) 15. Murphy, K.P., Weiss, Y.: The factored frontier algorithm for approximate inference in DBNs. In: Proceedings of the 17th Conference in Uncertainty in Artiﬁcial Intelligence, San Francisco, CA, USA, pp. 378–385 (2001) 16. Supplementary Materials, http://www.comp.nus.edu.sg/~ rpsysbio/cmsb09 17. Ammann, H.: Ordinary Diﬀerential Equations: An Introduction to Nonlinear Analysis. Walter de Gruyter, Berlin (1990) 18. Durrett, R.: Probability: Theory and Examples. Duxbury Press (2004) 19. Nunez, L.M.: On the relationship between temporal Bayes networks and Markov chains. Master’s thesis, Brown University (1989) 20. Kholodenko, B.N.: Untangling the signalling wires. Nat. Cell Biol. 9(3), 247–249 (2007) 21. Banga, J.R.: Optimization in computational systems biology. BMC Syst. Biol. 2(47), 1–7 (2008) 22. Gutenkunst, R.N., Waterfall, J.J., Casey, F.P., Brown, K.S., Myers, C.R., Sethna, J.P.: Universally sloppy parameter sensitivities in systems biology. PLoS Comput. Biol. 3(10), 1871–1878 (2007) 23. Hooke, R., Jeeves, T.A.: “Direct search” solution of numerical and statistical problems. J. ACM. 8(2), 212–229 (1961) 24. Hoops, S., Sahle, S., Gauges, R., Lee, C., Pahle, J., Simus, N., Singhal, M., Xu, L., Mendes, P., Kummer, U.: COPASI  a COmplex PAthway SImulator. Bioinformatics 22(24), 3067–3074 (2006) 25. Cho, K.H., Shin, S.Y., Kolch, W., Wolkenhauer, O.: Experimental design in systems biology, based on parameter sensitivity analysis using a monte carlo method: A case study for the TNFαmediated NFκB signal transduction pathway. Simulation 79(12), 726–739 (2003)
A Reduction of Logical Regulatory Graphs Preserving Essential Dynamical Properties Aur´elien Naldi1 , Elisabeth Remy2 , Denis Thieﬀry1,3 , and Claudine Chaouiya1,4 1
TAGC Inserm U928  Universit´e de la M´editerran´ee, Marseille, France 2 IML UMR 6206, Marseille, France 3 CONTRAINTES, INRIA Paris Rocquencourt, France 4 IGC, Instituto Gulbenkian de Ciˆencia, Oeiras, Portugal
Abstract. To cope with the increasing complexity of regulatory networks, we deﬁne a reduction method for multivalued logical models. Starting with a detailed model, this method enables the computation of a reduced model by iteratively “hiding” regulatory components. To keep a consistent behaviour, the logical rules associated with the targets of each hidden node are actualised to account for the (indirect) eﬀects of its regulators. The construction of reduced models ensures the preservation of a number of dynamical properties of the original model. In particular, stable states and more complex attractors are conserved. More generally, we focus on the relationship between the attractor conﬁguration of the original model and that of the reduced model, along with the issue of attractor reachability. The power of the reduction method is illustrated by its application to a multivalued model of the segmentpolarity network Controlling segmentation in the ﬂy Drosophila melanogaster. Keywords: Regulatory networks, logical modelling, model reduction, decision diagrams, regulatory circuits, stable states, complex attractors, Drosophila development, segmentation.
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Introduction
Biological data generation and integration eﬀorts result in the delineation of ever more comprehensive and complex regulatory networks involved in the control of numerous processes. Consequently, current modelling and analysis approaches are reaching their limits in terms of the number and variety of components and interactions that can be eﬃciently considered. This is true for quantitative frameworks (e.g., diﬀerential or stochastic models), as well as for qualitative approaches. Indeed, although logical modelling enables to handle networks comprising relatively large numbers of components (see e.g. [1,2]), the size of the state space grows exponentially with the number of regulatory nodes. P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 266–280, 2009. c SpringerVerlag Berlin Heidelberg 2009
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One way to handle this problem consists in developing compositional approaches to compute the dynamical properties of comprehensive networks, relying on the knowledge of the properties of simpler subsystems or modules. A complementary approach consists in reducing large systems, by focusing on the most relevant components and redeﬁning their interactions in order to preserve relevant dynamical properties (e.g. stable states). Most often, modellers intuitively and manually reduce regulatory networks to address speciﬁc questions. Such empirical reductions have several drawbacks: (i) the process is error prone and limited to relatively simple cases; (ii) the maintenance of diﬀerent versions of a model (complete and reduced) is cumbersome; (iii) storing the sole reduced model leads to the loss of relevant biological information. These considerations led us to develop a reliable, automated reduction method in the context of a logical modelling framework. In this respect, we lean on the software GINsim, which facilitates the deﬁnition of comprehensive logical regulatory graphs, as well as the analysis of their dynamical properties [3,4]. Established on ﬁrm mathematical bases, our reduction method allows the user to select nodes to be made implicit and to perform dynamical analyses on reduced model versions, which preserve relevant topological and dynamical properties. The paper is organised as follows. Section 2 recalls the deﬁnitions of logical regulatory graphs and of the associated state transition graphs. Next, the reduction method is deﬁned in Section 3. Relationships between the dynamical behaviour of the original model and that of the reduced model are delineated in Section 4. A multivalued logical model of the segmentpolarity network is then used to demonstrate the power of the proposed reduction method in Section 5. The paper ends with conclusions and further prospects. All models presented in this paper can be opened, edited, simulated, and analysed with GINsim, which implements the logical formalism and the reduction method presented here.
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Logical Modelling of Regulatory Networks
Our modelling approach leans on the generalised logical formalism initially developed by R. Thomas et al. [5,6,3]. In this context, a regulatory network and its dynamics are both represented in terms of oriented graphs. 2.1
Regulatory Graphs
Definition 1. A logical regulatory graph (LRG) is a directed labelled multigraph R = (G, Max, Γ, Θ, K) where, – G = {g1 , . . . , gN } is the set of nodes, representing regulatory components. – Max : G → N∗ associates a maximum level Max(gi ) = Maxi to node gi . The current level of gi , denoted xi , takes its values in Di = {0, . . . , Maxi }. – Γ is the set of arcs, deﬁned as a ﬁnite multiset of ordered pairs of elements of G representing regulatory interactions. If Maxi > 1, gi may have diﬀerent
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eﬀects onto a component gj , depending on level xi . Hence, the arc connecting gi to gj may be a multiarc encompassing diﬀerent interactions. The multiplicity of the arc (gi , gj ) (i.e. the number of its constitutive interactions), is denoted mi,j (1 ≤ mi,j ≤ Maxi ). Loops (even multiloops) are allowed: an arc (gi , gi ) denotes an autoregulation of gi . For each gj ∈ G, Reg(j) denotes the set of its regulators: gi ∈ Reg(j) if and only if (gi , gj ) ∈ Γ . – Θ is a labelling function, which associates a threshold to each element of Γ . More precisely, θi,j,k is associated to the k th interaction between gi and gj (denoted (gi , gj , θi,j,k ), k ∈ {1, . . . , mi,j }), with 1 ≤ θi,j,1 < · · · < θi,j,mi,j ≤ Maxi . This interaction is active, when xi , the level of its source gi , lays between the threshold of this interaction and that of the next interaction: θi,j,k ≤ xi < θi,j,k+1 (by convention, θi,j,mi,j +1 = Maxi + 1). – K = (K1 , . . . , KN ) deﬁnes the logical rules attached to the nodes specifying their behaviours: each Ki is a multivalued logical function that gives the target value of gi , depending on the levels of the regulators acting on gi : ⎛ ⎞ Ki : ⎝ Dj ⎠ → {0, . . . , Maxi }. gj ∈G
The logical function Ki can be equivalently deﬁned on the set gj ∈Reg(i) Dj , giving the target value of gi depending on the current levels of its regulators. Figure 1 illustrates this deﬁnition of a logical regulatory graph. In the following, when no confusion is possible, we will use i to denote gi .
G = {g1 , g2 , g3 , g4 } Max1 = Max2 = Max4 = 1 Max3 = 2 D1 = D2 = D4 = {0, 1}, D3 = {0, 1, 2} Reg(2) = {g1 , g3 } θ3,2,1 = 2
Fig. 1. Example of logical regulatory graph. Left: graphical representation of a LRG. Blunt arrows depict inhibitions while normal arrows depict activations (this is only a graphical convention, since the logical functions encode the regulatory eﬀects). The rectangular node g3 is ternary, whereas the others nodes are Boolean. The thresholds of all interactions are set to 1, except that of (g3 , g2 ), which is set to 2. Right: illustration of the notations of Deﬁnition 1. Examples of logical functions Ki are displayed in Figure 2 for the same model.
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State Transition Graphs
We represent the dynamical behaviour of a LRG in terms of a state transition graph, deﬁned as follows.
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Definition 2. Given a LRG R = (G, Max, Γ, Θ, K), its associated full state transition graph (STG) E = (S, T ) is a directed graph, where: – S = Πi∈G Di is the state space, a state of the system being a vector x = (xi )i=1,...,N , with xi ∈ Di , ∀i ∈ G, – T ⊂ S 2 is the set of transitions deﬁned as follows: (x, y) ∈ T if and only if ∃i ∈ G such that: xi = Ki (x), Ki (x)−xi y = x + Δi (x).ei , where Δi (x) = K and ei is the canonical vector i (x)−xi  i i in S (ei = 1 and ej = 0, ∀j ∈ G, j = i). Here Δi (x) gives the sign of the update of i (increase or decrease). One can also consider a state transition graph related to an initial (set of) condition(s). It is then a subgraph of the full STG. When analysing the behaviour of a LRG, we mainly focus on attractors, which represent asymptotic dynamical properties. Given a STG, attractors are its terminal strongly connected components, classiﬁed as: – stable states: reduced to a unique terminal node, – cyclic attractors: terminal elementary (oriented) cycles, – complex attractors: other terminal strongly connected components (i.e. involving intertwined cycles). Cyclic and complex attractors will be called nontrivial attractors. In what follows, LRGs are assumed consistent, i.e. all interactions are eﬀective and autoregulations functional, meaning that all interactions have a dynamical role and could be recovered from the logical functions K (see further explanations in the Appendix A).
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Logical Regulatory Graph Reduction
This section presents the principles underlying the reduction of a regulatory graph and then deﬁnes the new model, called reduced model. In what follows, we consider a reduction consisting of the removal of a single regulatory component (making it implicit). The generalisation to a reduction encompassing a set of nodes is obtained by iterating the corresponding onenode reductions. However, the ordering of a sequence of onenode reductions may have an impact on the resulting reduced model (see Appendix B). Here, we aim at deﬁning a reduction method, which preserves, as much as possible, the dynamical properties of the original model. The underlying principle is already intuitively applied by modellers when they make regulatory nodes implicit in their networks. The removal of a node r basically consists in connecting directly its regulators to its targets, which logical functions are thus revised. In the revised logical functions, the eﬀect of r at a given value xr is conveyed by the values of the regulators leading r to xr . In other words, we consider the update of the removed component as a fast process, which is performed before anything else.
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Following this principle, it is impossible to remove an autoregulated component since ﬁxed values of its other regulators may not lead to a unique target value. Thus, the removal of an autoregulated component implies additional decisions, impeding the deﬁnition of a systematic procedure. In the following, we will require that autoregulated components should not be removed. To properly implement an algorithm producing the reduced model, we need further notations to manipulate the logical functions. Given a regulatory graph R = (G, Max, Γ, Θ, K) and a node i ∈ G, we denote: {l}
– xi (l ∈ Di ) the Boolean variable with value 1 when xi = l, 0 otherwise. – xSi the Boolean variable that is true if xi ∈ S, false otherwise. Hence xSi is deﬁned by, {l} xSi xi , S ⊆ Di . l∈S i Note that x∅i is always false and xD i always true. – For all v ∈ Di , the logical function Kiv that gives the conditions under which the target value of node i is v. This function is deﬁned as follows: Kiv = Cin , (1)
n=1,...p
S where Cin are conjunctive clauses Cin = j∈Reg(i) xj j,i,n , where Sj,i,n ⊆ Dj . n Each clause Ci deﬁnes a situation (i.e. sets of combinations of incoming interactions acting upon i) for which the target value of i is v. In Equation (1), each clause Cin deﬁnes a subset of S, D = Πj∈G Sj,i,n (with Sj,i,n = Dj , ∀j ∈ / Reg(i)), such that for all x ∈ D, Ki (x) = v. Hence, Equation (1) deﬁnes a set of cubes in the state space S, where the target value of i is v. Definition 3. Given a LRG R = (G, Max, Γ, Θ, K), the reduced LRG Rr = (G r , Maxr , Γ r , Θr , Kr ) obtained by removing a nonautoregulated component r ∈ G is deﬁned as follows: – G r = G \ {r}. – Maxr : G r → N∗ , s.t. ∀i ∈ G r Maxr (i) = Maxi . – For all i ∈ G r , and for all v ∈ Di , the logical function Kirv is deﬁned as follows. Consider Kiv = n=1,...p Cin , the disjunctive form of Kiv , as deﬁned previously. For all n = 1, . . . p (i.e. for each clause Cin ), let deﬁne Firn as: ⎛ ⎞ ⎛ ⎞ S Firn = ⎝ Krw ⎠ ∧ ⎝ xj j,i,n ⎠ w∈Sr,i,n
j∈Reg(i)\{r}
Then Kirv = n=1,...p Firn . – Γ r and Θr are deduced from Kr ; for all i ∈ G r , j ∈ G r , mri,j = 11i,j,v , v∈[1,Maxi ]
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Fig. 2. Reduction in terms of MDDs. Left: the same LRG as in Figure 1, where g4 (greyedout) is selected for removal. Logical functions for g1 and g4 are shown on the right, along with their MDD representations. Right: the reduced LRG after removal of g4 , along with the resulting logical function for g1 . In the MDDs, internal nodes are labelled with the associated variable (xi ), whereas leaves represent the value of the logical functions. Children of internal nodes are ordered from left to right: the leftmost (resp. rightmost) child is the root of the subdiagram corresponding to the case xi = 0 (resp. xi = Maxi ).
where 11i,j,v = 1 if it exists x ∈ S such that xi = v−1 and Kjr (x) = Kjr (x+ei ). r r Then (i, j) ∈ Γ if mi,j > 0 (and the multiplicity of (i, j) in Γ r is given by mri,j ). Finally, the ordered set of values v such that 11i,j,v = 1 deﬁnes the r thresholds θi,j,k (k = 1, . . . , mri,j ). The logical function Kirv is deduced from the logical function Kiv by replacing, in each clause, literals xSr by the formulae giving the conditions under which the target value of r is in S (remark that this deﬁnition may not give Kirv in a proper disjunctive form). Note that if Cin does not depend on r (i.e. r ∈ / Reg(i)) then Sr,i,n = Dr and Firn = Cin for all n, therefore Kirv = Kiv . The set of arcs veriﬁes: Γ r ⊆ {(i, j) ∈ G r × G r , s.t. (i, r), (r, j) ∈ Γ or (i, j) ∈ Γ }. In practice, the construction of the new logical function is performed using Reduced Ordered Multivalued Decision Diagrams (ROMDDs or MDDs for short). Decision diagrams are rooted directed acyclic graphs, widely used to represent logical functions (see e.g. [7,8]). In these diagrams, internal nodes are labelled with decision variables and have one child per value, while leaves represent the values of the function. Decision variables are ordered: each internal node has a rank and the subdiagrams rooted by the children of a node of rank i do not contain internal nodes of rank j ≤ i. In [9] we used MDDs to represent the logical functions Ki . In this context, decision variables are the levels of the components of the model. For the sake of simplicity, we consider that the ordering of the MDD variables is the same as that of the LRG components. Given the MDD representation of Ki and a state x, a unique path from the root of the MDD to one of its leaves is deﬁned. Along this path, the child chosen for each nonterminal node is labelled with the value of the corresponding variable in state
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x. The terminal node reached through this path gives the value of Ki (x). Each clause of Kiv corresponds to a path leading to a leaf valued v. To compute the MDD representing Kir , we deﬁne the recursive algorithm given in Appendix C and illustrated in Figure 2.
4
Dynamics of the Reduced Model
In this section, the dynamical behaviour of a reduced LRG (as speciﬁed in Definition 3) is compared to that of the original LRG. In particular, we show that the reduction preserves existing attractors and does not add any spurious path. Let E = (S, T ) be the full state transition graph of R = (G, Max, Γ, Θ, K) and r ∈ G a node not autoregulated. Let E r = (S r , T r ) be the full STG of Rr = (G r , Maxr , Γ r , Θr , Kr ), the LRG obtained after the removal of r from G. Consider the projection πr : S → S r such that, ∀i ∈ G r , ∀x ∈ S, (πr (x))i = xi , and the equivalence relation on S: ∀x, y ∈ S, x ∼r y iﬀ πr (x) = πr (y). We denote [x]∼r the equivalence class: [x]∼r = {y ∈ S s.t. y ∼r x}. The class [x]∼r contains all states of S that diﬀer only by their rth component, i.e. the (Maxr + 1) states {xi ∈ S, i = 0, . . . , Maxr }, such that xi ∼r x and xir = i. Because r is not autoregulated, ∀xi ∈ [x]∼r , Kr (xi ) = Kr (x). This implies that: – (xi , xi+1 ) ∈ T , for all i < Kr (x), – (xi , xi−1 ) ∈ T , for all i > Kr (x), – (xKr (x) , xi ) ∈ / T , for all i. Hence, for all x ∈ S, there exists a path in S from x to xKr (x) , which is the representative state of [x]∼r . Definition 4. x ∈ S is the representative state of an equivalence class for ∼r iﬀ xr = Kr (x) . We can then deﬁne the retrieval function sr : S r → S such that, ∀z ∈ S r , (sr (z))i = zi , for all i ∈ G \ {r}, (sr (z))r = Kr (x), with x such that πr (x) = z. In other words, sr (z) is the representative state of the equivalence class projected on z (see Figure 3). Relying on this, we can introduce an alternative deﬁnition of the logical functions in the reduced LRG: ∀i ∈ G r , Kir : S r → Di is deﬁned as Kir (z) = Ki (sr (z)). Note that if (r, i) ∈ Γ (i.e. r is not a regulator of i), Kir (πr (x)) = Ki (x). Remark 1. It follows from their deﬁnitions that functions πr and sr verify: 1. 2. 3. 4.
πr ◦ sr is the identity function. For any x ∈ S, (sr ◦ πr (x)) ∼r x. If x ∈ S is a representative state, then, sr ◦ πr (x) = x. For any z ∈ S r , Kr (z) = πr (K(sr (z))) ; indeed, ∀x ∈ S, ∀i ∈ G r , Kir (πr (x)) = Ki (sr ◦ πr (x)).
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Fig. 3. Dynamical behaviour of the reduced model given in Figure 2, before and after removal of the ternary node g3 . Left: State transition graph (STG), partitioned into four equivalence classes for g3 . Each equivalence class contains 3 states; its representative state is greyed out and internal transitions are dashed. Right: STG of the reduced model, each state corresponding to an equivalence class of the original STG. After the reduction, the stable state 102 is projected on 10 and all transitions are preserved except the one from the second equivalence class to the third one. This results in the isolation of the nonterminal strongly connected component involving the ﬁrst two equivalence classes of the original STG, hence generating the attractor (01, 00).
The following lemma establishes the relationships between transitions in E and E r . Lemma 1. 1. Let z, z ∈ S r . (z, z ) ∈ T r =⇒ ∃x ∈ S s.t. πr (x) = z and (sr (z), x) ∈ T . 2. Let x, y ∈ S. If x is a representative state, then (x, y) ∈ T
Proof. Recall that Δi (x) = denote:
Δri (z) =
=⇒
Ki (x)−xi Ki (x)−xi  .
(πr (x), πr (y)) ∈ T r . For z ∈ S r s.t. zi = Kir (z), we similarly
Kir (z) − zi Ki (sr (z)) − (sr (z))i = = Δi (sr (z)) . Kir (z) − zi  Ki (sr (z)) − (sr (z))i 
1. Consider z, z ∈ S r such that (z, z ) ∈ T r . Then ∃i = r s.t. Kir (z) = zi , and r i r z = z + Δi (z) e . By deﬁnition, Ki (z) = Ki (sr (z)) = (sr (z))i = zi . This implies that (sr (z), x) ∈ T with x ∈ S and x = sr (z) + Δi (sr (z)) ei , and then πr (x) = z . 2. Consider x, y ∈ S such that Kr (x) = xr . The hypothesis (x, y) ∈ T implies that ∃i ∈ G, i = r s.t. Ki (x) = xi , and y = x + Δi (x) ei .
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We have Kir (πr (x)) = Ki (x) (since x is a representative state), and xi = (πr (x))i , since i = r. So, Kir (πr (x)) = (πr (x))i , and then ∃z ∈ S r s.t. r (πr (x), z) ∈ T , with z = πr (x) + Δri (πr (x)) ei = πr (x) + Δi (sr ◦ πr (x)) ei = πr (y) .
r
The ﬁrst item of Lemma 1 states that any transition in T corresponds to at least one transition in T . Clearly, the reverse is not true. The second item of the lemma gives a condition under which transitions are preserved from T to T r . Of course, it is important to know which transitions are lost through the reduction. Definition 5. The reduction preserves a transition (x, y) ∈ T if (πr (x), πr (y)) ∈ T r , or πr (x) = πr (y). The reduction preserves a path (s1 , . . . , sn ) ∈ E if all its transitions are preserved. In other words, a path (s1 , . . . , sn ) in E is preserved if the reduction preserves the transitions between equivalence classes, in the required order. The following property characterises the transitions that are not preserved by the reduction. Property 1. A transition (x, y) ∈ T is not preserved by the reduction if and only if the three following conditions are satisﬁed: 1. x is not a representative state, 2. y ∈ [x]∼r (⇒ ∃i = r s.t. yi = xi ), 3. Δi (x) = Δi (sr ◦ πr (x)) . The last condition means that there is no call for updating i in the same direction in state sr ◦ πr (x). Proof. Consider a transition (x, y) ∈ T , which satisﬁes the three conditions. Suppose that (x, y) is preserved by the reduction, then (πr (x), πr (y)) ∈ T r (the case πr (x) = πr (y) is not possible because of the second condition). This means that there exists j = r s.t. (πr (x))j = (πr (y))j , and (πr (x))k = (πr (y))k for any k = j. With Condition 2 and by deﬁnition of πr , we deduce that j = i. Moreover, we know that: πr (y) = πr (x) + Δri (πr (x)) ei = πr (x) + Δi (sr ◦ πr (x)) ei . Finally, y = x + Δi (x) ei , and, as yi = (πr (y))i , we have Δi (x) = Δi (sr ◦ πr (x)). This contradicts Condition 3. Hence, (x, y) is not preserved by the reduction. Conversely, let (x, y) ∈ T be a transition not preserved by the reduction. – Condition 1 is satisﬁed by the second item of Lemma 1. – Condition 2 is satisﬁed because y ∈ [x]∼r ⇒ πr (x) = πr (y) ⇒ (x, y) preserved, hence a contradiction. – We know that y = x+Δi (x) ei . As Kr (πr (x)) = πr (K(sr ◦πr (x))) (cf. Remark 1), Kir (πr (x)) = (πr (K(sr ◦ πr (x))))i = Ki (sr ◦ πr (x)) = xi = (πr (x))i .
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Hence, there exists z ∈ S r s.t. (πr (x), z) ∈ T r with zi = πr (x) + Δri (πr (x)) ei = πr (x) + Δi (sr ◦ πr (x)) ei = πr (x) + Δi (x) ei . Consequently, πr (y) = z and (x, y) is preserved, hence a contradiction.
Given C, a set of states in S, we denote πr (C) = {πr (x), x ∈ C}. Given C ,
a set of states in S r , we denote sr (C ) = {sr (z), z ∈ C }. Note that πr (C) may contain less elements than C, and that sr (C ) contains only representative states. The following results relate attractors in E and E r . Proofs are provided in Appendix D and E. Theorem 1. Consider a LRG R = (G, Max, Γ, Θ, K) and Rr the reduced LRG. Let E (resp. E r ) be the full STG of R (resp. of Rr ), then: 1. Stable states in E and E r verify: – x stable state in E =⇒ πr (x) stable state in E r . Furthermore no other stable state is projected on πr (x), – z stable state in E r =⇒ sr (z) stable state in E. Hence, the number of stable states is conserved by the reduction. 2. If (s1 , . . . sn ) is a cyclic attractor in E, then (πr (s1 ), . . . πr (sn )) is a cyclic attractor in E r . 3. If C is a complex attractor in E, z ∈ πr (C) and (z, z ) ∈ T r , then z ∈ πr (C). As a consequence, πr (C) contains at least one nontrivial attractor in E r . Theorem 1 characterises the dynamical properties conserved by the reduction. Going further, it is possible to identify the situations leading to the generation of additional nontrivial attractors. A nontrivial attractor in the reduced STG corresponds to a (part of a) strongly connected component of the original STG. This SCC is itself a nontrivial attractor or involves outgoing transitions all in conﬂict with transitions concerning the removed component. In other words, we can fully characterise the set of states in the original STG giving rise to a nontrivial attractor in the reduced dynamics. Interestingly, this set corresponds to transient oscillatory behaviour from which the system cannot escape provided that updates of the removed component are always faster than other concurrent changes. This is formalised by Theorem 2 in Appendix E.
5
Application: Segment Polarity
We demonstrate the power and ﬂexibility of our reduction method through its application to the segmentpolarity network, which plays a key role in the segmentation of the ﬂy embryo. This system has been thoroughly analysed by developmental geneticists and has been already modelled using continuous [10,11,12] and logical approaches [13,14,15]. However, all these studies involved important simpliﬁcations of the network, particularly so as a proper modelling of
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Fig. 4. Logical model of the segment polarity network for two cells, based on [15]. Ellipsoid and rectangular nodes denote Boolean and ternary components, respectively. The two cellular networks have been properly connected to take into account Wg and Hh diﬀusion, as well as Hh sequestration by Ptc, as in [16]. The anterior cell contains the extended version of the model, where greyedout components will be removed, leading to the model on the right. Dashed arrows denote indirect interactions resulting from this reduction. Greyedout components in the posterior cell are candidates for further reduction.
its behaviour requires the chaining of several identical networks to account for intercellular interactions through Wingless (Wg) and Hedgehog (Hh) signalling. Describing the most complete model to date, [15] had to discard various components known to play important roles in Wg and Hh signalling to keep dynamical simulations and analyses computationally tractable for up to six cells. Here, we propose a logical model based on their full description of the segment polarity network. The resulting regulatory graph encompasses 18 components and 31 regulatory interactions (left part of Figure 4). In order to model the intercellular interactions involved in the formation of segment boundaries, we have to connect neighbouring cells (along the anteriorposterior axis) through Wg and Hh signalling. Wg is known to bind its receptor, Frizzled (Fz), only at very short range, amounting here to neighbouring cells. This can be represented by positive arcs linking each Wg node to Fz nodes of neighbouring cells. In contrast, Hh is able to reach more distant cells, but can be sequestered by its receptor Patched (Ptc). Similar interactions have been modelled in [16] in terms of positive arcs between Hh nodes in neighbouring cells (diﬀusion) and negative arcs from Ptc onto the Hh node of neighbouring cells (sequestering). Figure 4 illustrates the intercellular network obtained after coupling two cells and reducing one of the cellular subnetworks down to nine components. The reduction method described above can be advantageously applied to ease the identiﬁcation of all attractors of such intercellular models (S´ anchez et al.
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Table 1. Dynamical characteristics of diﬀerent reduced models derived from that of Figure 4 (involving 2 x 9 nodes after applying the same reduction to both cells). The number of reachable states decreases drastically with the number of considered nodes. Note that the three stable states remain reachable for all reductions listed, but the last one (removal of Slp). LRG size Removed components Number of reached states Reached stable states 2x9 – > 106 TT, WE, EW 2x7 Fz,Ptc 12476 TT, WE, EW 2x6 Fz,Ptc,Nkd 1625 TT, WE, EW 2x8 Slp 11350 TT, WE
considered six cells). The modeller can select the sets of nodes to discard from the network, depending on biological considerations (e.g. diﬀerent time scales, speciﬁc mutations, etc.). In a ﬁrst step, it is reasonable to conserve transcription factors and components involved in intercellular communications: Wg, Hh and their receptors (Fz and Ptc). However, since the transcription factor Cubitus interruptus is represented by three nodes here (full length immature Ci protein, activator Ciact and repressor Cirep forms), we choose to retain only the two nodes corresponding to active regulatory forms. These choices correspond to the removal of the greyedout components in the left part of Figure 4. The reduced model involves half of the nodes of the original one, which amounts to a much higher reduction of the number of possible states, as this grows exponentially with the number of regulatory nodes. The resulting regulatory graph (Figure 4, right) remains easy to grasp as it reasonably unfolds most intracellular and intercellular regulatory pathways. As we shall see, this logical model can be further reduced to facilitate analyses encompassing more cells. For proper logical rules (cf. [15] and supplementary material), one can check that the detailed and the reduced twocells models have exactly three stables states (as predicted by Theorem 1). These multicellular stable states combine three types of cellular states: a Wg expressing state (denoted W), an En expressing state (E), and a trivial state (T) expressing neither Wg, nor En. The three stable states for the two connected cells correspond to TT, WE and EW cell combinations reported by S´ anchez et al. All three stable states are reachable from biologically relevant initial conditions (signiﬁcant amounts of Wg and Slp in the anterior cell, signiﬁcant amount of En in the posterior cell), provided as an outcome of the activity of the pairrule system, cf. [17,15]. However, the size of the corresponding state transition graph still impedes detailed dynamical analyses (see Table 1). As shown in Table 1, the removal of Fz, Ptc and Nkd drastically reduces the number of reached states without changing the reachability of the three stable states from the considered initial state. However, the sole removal of Slp impedes the reachability of the stable state with inverted Wg and En expressing cells. It also suppresses the functionality contexts of all negative circuits [9,15], implying that the state transition graph does not contain any cyclic attractor. Indeed, after further reduction to three nodes per cell (Wg, En and Hh), we were able
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to check the absence of nontrivial attractors in the full STG. As the reduction cannot delete existing nontrivial attractors (see Theorem 1), this implies that all attractors of the original model are stable states.
6
Conclusions and Prospects
We have deﬁned a reduction method that can be applied to multivalued logical models while preserving important dynamical properties. In particular, all attractors of the original dynamics have a counterpart in the dynamics of the reduced model. Furthermore, trajectories in the reduced model can be formally related to trajectories in the original one. This enables to infer the existence of paths in the dynamics of a detailed model whenever it is possible to show (by simulation and graph analysis) that paths exist between the corresponding states in a reduced version of the model. However, the reverse is not true. Indeed, a reduction can lead to the loss of reachability properties. Whenever several asynchronous component updates are possible at a given state, the elimination of one of the updated components amounts to consider it as ”faster” than the concurrent ones, leading to the possible exclusion of some transitions in the reduced STG. Such reductions relate to the delineation of speciﬁc priority class conﬁgurations [18]. One particular feature of the reduction method deﬁned here is that the removal of (functional) autoregulated components is forbidden. This rule is related to previous work on the dynamical roles of the regulatory circuits. Indeed, it has been recently proven in the discrete framework that positive regulatory circuits are necessary to generated multiple attractors, whereas negative circuits are necessary to generate cyclic attractors (cf. [19] and references therein). At least in the discrete framework, these properties depend only on the sign of the regulatory circuit, i.e. on the product of the sign of the involved interactions and not on their number. From a qualitative dynamical point of view, it is thus possible to reduce the number of components of a circuit down to a single autoregulated component, while keeping the corresponding property, as long as we conserve the sign of the circuit (along with some functionality constraints). Our formal presentation of the reduction method mainly focuses on the removal of a single component. However, iterating this process enables the removal of several components. This raises the question of the impact of the order in which reductions are performed. As shown in Appendix B, the removal of a component may be possible only after the prior removal of others. If we aim at removing as many components as possible, the ordering of removals may thus be crucial. Further work is needed to properly deﬁne optimal or maximal reductions for the general case. When the removal of a set of components is possible in several orders, we suspect that the dynamics of the resulting model does not depend on the order (work in progress). The worst case complexity of the algorithm for the reduction of a node r that regulates k targets is in O(md ), where m is the highest number of levels of the involved components and d is the depth of the MDDs representing the revised logical functions associated to the target nodes. In most cases, m ≤ 3 and d ≤ 5.
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Applying our reduction method to a detailed model of the segmentpolarity network, we were able to show the absence of nontrivial attractors in a state transition graph too large to be stored. As indicated for this application, the reduction method oﬀers a great ﬂexibility to the modeller. Biological arguments (e.g. information on relative reaction speeds) can be used to select sets of nodes for consistent model reduction. In the course of the dynamical analysis of complex networks (e.g. multicellular networks), further reduction can be performed to identify all attractors and check their reachability from speciﬁc initial states. To ease the maintenance of a detailed model along with its reduced versions, the GINsim implementation enables the user to deﬁne and record various reductions for the same reference model. In order to handle still larger and more complex networks, such reduction could be combined with algorithmic methods enabling the analysis of large state transition graphs ([20] and references therein), or yet with model checking techniques ([21] and references therein). Supplementary Materials. GINsim can be downloaded from http:// gin.univmrs.fr/GINsim. The Appendix, and the models, are available at the following URL: http://gin.univmrs.fr/GINsim/publications/ naldi2009.html. Acknowledgments. A.N. has been supported by PhD grant from the French Ministry of Research and Technology. C.C. acknowledges the support provided by the Calouste Gulbenkian Foundation. This work was further supported by research grants from the French National Agency (project ANR08SYSC003), from EU FP7 (APOSYS Project), and by the Belgian Science Policy Oﬃce (IAP BioMaGNet).
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On the Use of Stochastic Petri Nets in the Analysis of Signal Transduction Pathways for Angiogenesis Process Lucia Napione, Daniele Manini, Francesca Cordero, Andr´as Horv´ath, Andrea Picco, Massimiliano De Pierro, Simona Pavan, Matteo Sereno, Andrea Veglio, Federico Bussolino, and Gianfranco Balbo 1
Department of Computer Science, University of Torino, Torino, Italy Institute for Cancer Research and Treatment, Candiolo (TO), Italy Department of Clinical and Biological Sciences, University of Torino, Torino, Italy 4 Department of Oncological Sciences, University of Torino, Torino, Italy 2
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Abstract. In this paper we consider the modeling of a selected portion of signal transduction events involved in the angiogenesis process. The detailed model of this process contains a large number of parameters and the data available from wetlab experiments are not sufficient to obtain reliable estimates for all of them. To overcome this problem, we suggest ways to simplify the detailed representation that result in models with a smaller number of parameters still capturing the overall behaviour of the detailed one. Starting from a detailed stochastic Petri net (SPN) model that accounts for all the reactions of the signal transduction cascade, using structural properties combined with the knowledge of the biological phenomena, we propose a set of model reductions.
1 Introduction Formal modeling is a central theme in systems biology in which mathematical modeling and simulation can play an important role. The Petri net (PN) formalism [18] is a framework that allows the construction of a precise and clear representation of biological systems based on solid mathematical foundations. This formalism permits the study of qualitative properties related to the structure of the model (e.g., the structure of a biological pathway). The variant of PNs, called Stochastic Petri Nets (SPNs) [16,15,2] and characterized by the addition of timing and/or stochastic information, can be used for quantitative analysis (e.g., analysis that involve the rates in biochemical reactions). PNs have been first proposed for the representation of biological pathways by Reddy et al [17]. Since their introduction, many other researchers constructed PN models of biological pathways [11] with the aim of using their representations to obtain qualitative information about the behavior of these systems, mostly via simulation [12,9]. The interaction of qualitative and quantitative analysis is necessary to check a model for consistency and correctness; following this idea, Heiner et al [10] proposed a methodology to develop and analyze large biological models in a stepwise manner. In this paper we present our experience in modeling signal transduction pathways for the angiogenesis process using SPNs. The general goal is to analyze the temporal dynamics of a few relevant biological products and this requires to build and parameterize P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 281–295, 2009. c SpringerVerlag Berlin Heidelberg 2009
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the model of the phenomenon under study. A detailed model is built by biologists and then the parameters are estimated on the basis of data obtained by wetlab experiments. It is often the case however that the amount of available wetlab data is not sufficient to have reliable estimates of the many parameters involved in the model. The key contribution of this paper aims at alleviating this problem by providing a simplification process which transforms the detailed model into a simpler one with less parameters. The proposed simplification process is guided by qualitative properties together with knowledge on the phenomenon under study and it is validated by comparing the quantitative properties of the detailed and simplified models. Moreover, this process represents the basis of the development of arguments useful for identifying both critical complexes and interactions that play a crucial role in the biochemical system under study. With respect to the framework proposed by Gilbert et al [8], where the main idea is to illustrate the complementarity among the three different ways of modeling biochemical network , i.e. qualitative, stochastic and continuous, here we focus our attention on the definition and robustness of the simplification process to limit the complexity of the model. Techniques that can be seen as simplification procedures have already been published in the literature. See, for example, [4,3] where approximate analysis methods based on aggregation of states are proposed. The goal of these techniques is however different from ours, since they aim at reducing the complexity of the analysis of the model and not the difficulty of its parametrization. Indeed, they result in simpler models in which the number of parameters is identical to that of the original one. The paper is organized as follows. Section 2 provides an overview of PNs and SPNs and of their use in biochemical systems. Section 3.1 describes the angiogenesis case study. Section 3.2 presents the approach we followed to build the SPNs and Section 3.3 shows the formal and biological rules used in the simplification process as well as the resulting SPNs. The quantitative analysis performed in order to verify the mathematical robustness of the simplified model is proposed in Section 3.4. We conclude with a discussion and an outlook of future works in Section 4.
2 Modeling Formalism and Solution Techniques The descriptions commonly applied in biology, where the relations among components are expressed by biochemical reactions, or by interactions of genes as well as by cell population interactions, are easy to transform into PNs in which places correspond to genes/proteins/compounds (substrates) and transitions to their interactions. 2.1 Petri Net Representation for Biochemical Entities Interactions PNs are a graphical language for the formal description of distributed systems with concurrency and synchronization. PNs are bipartite graphs with two types of nodes, namely places and transitions, connected by directed arcs. The state of the system is given by the distribution of tokens over the places of the net. The dynamics of the model (starting from an initial marking) is captured by state changes due to firing of transitions and by the consequent movement of tokens over the places.
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Definition 1. [PN  syntax ]A Petri Net graph is a tuple (P, T, W, m0 ), where P is a finite set of places; T is a finite set of transitions; P and T are such that P ∩ T = ∅; W : (P × T ) ∪ (T × P ) → IN defines the arcs of the net and assigns to each of them a multiplicity; – m0 is the initial marking which associates with each place a number of tokens. – – – –
When applied in systems biology, places represent biochemical entities (enzymes, compounds, etc.) and transitions represent the interactions among entities [17]. The quantities of the entities are represented by tokens in the places. The biological system we consider consists of biochemical reactions similar to those reported in Fig. 1a where we show the PN representation scheme we adopted to describe all reactions of this type. Fig. 1b represents the state evolution due to firing of transition k53 . P ip3
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2.2 Analysis Techniques Based on Structural Properties The PN graph inspection can provide several functional properties of the model, whose validity is true independently of the initial state of the system: such properties are, for instance, the boundedness and the existence of structural deadlocks and traps [18,2]. In deriving such kinds of information an important role is played by the so called net’s invariants. There exist two kinds of invariants: place invariants (Pinvariants) and transition invariants (Tinvariants) [18]. In this paper we deal with Pinvariants only. A Pinvariant is a weighted sum of tokens contained in a subset of places of the net that remains constant through the entire evolution of the model, starting from an initial marking. The subset of places used for computing the Pinvariant is the support (i.e., the set of nonzero components) of a Psemiflow f [14], which is a vector of nonnegative weights assigned to all the places of the net. A Psemiflow f is an integer and nonnegative solution of the matrix equation fC = 0, where C is the incidence matrix of the net, obtained by properly using the information provided by the flow relation W . The interpretation of a Pinvariant in a biological context, where tokens represent compounds, enzymes etc., is relatively simple: the places that support the semiflow f represent a portion of the PN where a given kind of correlated matter is preserved.
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2.3 Quantitative Temporal Analysis To study the temporal dynamics of a biological system it is natural to apply an extension of PNs that allows to introduce in the model temporal specifications. SPNs are time extensions of PNs in which exponentially distributed random delays (interpreted as durations of certain activities) are associated with the firings of the transitions. SPNs are qualitatively equivalent to PNs, meaning that for their structural analysis it is sufficient to disregard their time specifications. The temporal stochastic behaviour of an SPN is isomorphic to that of a continuous time Markov chain (CTMC). This CTMC can be built automatically from the description of the SPN and corresponds to the behaviour of the biological system described by the Master Chemical Equations [7]. This “stochastic approach” based on SPNs, adopts a discrete view of the quantity of the entities and sees their temporal behaviour as a random process. Another possibility is to adopt a “deterministic approach” in which the temporal behaviour of the entities is seen as a continuous and completely predictable process. In our context we make use also of the deterministic approach because it allows for faster and simpler evaluation of the simplification process we propose for SPNs. The deterministic approach translates the interactions into a set of coupled, firstorder, ordinary differential equations (ODE) with one equation per entity. These equations describe how the quantities of the entities change based on the speed and the structure of the interactions among reactants. Referring again to the reactions considered in Figure 1, the corresponding ODEs are dXP ip3 (t) = −k53 XP ip3 (t)XP ten (t) + k54 XP ip3:P ten (t), dt dXP ten (t) = −k53 XP ip3 (t)XP ten (t) + k54 XP ip3:P ten (t), dt dXP ip3:P ten (t) = k53 XP ip3 (t)XP ten (t) − k54 XP ip3:P ten (t) dt
where Xi (t) denotes the quantity of reactant i at time t. Having the ODEs and information on the initial amount of the different entities, numerical integration of the ODEs is applied to calculate the quantities at a given time instant.
3 A Stochastic Petri Nets Based Approach Applied to Signal Transduction Pathways for the Angiogenesis Process One main objective in systems biology is to model and analyze temporal dynamics of the phenomenon under study. By using SPNs as the formalism for the construction of the model, the analysis is performed in two steps: the first provides qualitative information on the structure of the model and the second investigates quantitative properties including statistical indices describing the temporal behavior of the system. Here we use this approach to study the angiogenesis process. 3.1 Biological Case Study Definition Angiogenesis, defined as the formation of new vessels from the existing ones, is a topic of great interest in all areas of human biology, particularly to scientists studying vascular development, vascular malformation and cancer biology. Angiogenesis is
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a complex process involving the activities of many growth factors and relative receptors, which trigger several signaling pathways resulting in different cellular responses. The Vascular Endothelial Growth Factor (VEGF) family proteins are widely regarded as the most important growth factors involved in angiogenesis. VEGFA, a member of VEGF family, has been most carefully studied and is thought to be of singular importance. VEGF receptor2 (KDR in humans) is thought to mediate most of VEGFA’s angiogenic functions, including cell proliferation, survival, and migration. Although the core components of the main KDRinduced pathways have been identified, the molecular mechanisms involved need to be characterized in fine details in order to better understand the flow of information. Indeed, a strong body of evidences indicates the presence of common adaptor/effector proteins involved in the survival and proliferation pathways induced by VEGFA/KDR axis, pointing out the difficulty to isolate a specific pathway and suggesting the presence of common nodes which contribute to create an intricate signaling network. In particular, the phosphorylated active receptor, indicated as KDR∗ , catalyzes phosphorylation of several intracellular substrates including the adaptor protein Gab1 [13,5]. The main pathway through which VEGFA induces cell proliferation involves the activation of P LCγ [19]. Activation of P LCγ promotes phosphatidylinositol 4,5bisphosphate (P ip2 ) hydrolysis giving rise to 1,2diacylglycerol (DAG). VEGFAinduced cell survival is dependent on the activity of P i3K [6]. The activated P i3K phosphorylates P ip2 generating phosphatidylinositol3,4,5triphosphate (P ip3 ). This recruits Akt to the membrane where it is activated trough phosphorylation. Activated Akt induces cell survival. Taking into account these notions, we wrote a system of biochemical reactions based on the available biological information together with further supposed mechanisms which could contribute to underline the presence of additional molecular nodes in the context of VEGFAinduced proliferation and survival pathways. 3.2 Model Construction In this section we discuss the approach we followed to represent the signal transduction cascade by SPN. Consider the detailed biological model depicted in Fig. 2. These reactions describe KDRproximal signaling events in the context of the survival and proliferation signal modules induced by receptor activation. In particular, reactions are split into four blocks. The First Block represents the earliest signaling events which include KDR∗ (we use the star to denote that proteins are active), Gab1, and P ip3 . The Second Block concerns the regeneration of P ip2 , a common substrate for the two signal modules that we are considering. In this block P ip2 recovery was considered to result from the contribution of Ptendependent dephosphorylation of P ip3 in combination with DAG catabolism (here recapitulated in the pseudoenzyme E). The Third Block includes the reactions describing the survival pathway triggered by the P I3K/Akt axis. The Fourth Block represents the proliferation pathway involving P LCγ activation. Using the reaction representations outlined in Section 2.1 and the GreatSPN tool [1] the SPN model of the angiogenesis process was built as illustrated in Fig. 3. Exploiting the block organization and the structure of the model we analyzed the biochemical reactions in order to identify possible pathways and subpathways that describe embedded behaviors of the complete model. We denoted the reactions by means of their kinetic constants.
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60 P ten:P ip2:P ip3 → P ten:P ip2 + P ip2 —————
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36 Kdr∗:P lc∗γ :P ip2 → Kdr∗:P lcγ + Dag —————
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39 Kdr∗:Gab1∗:P lcγ → Kdr∗:Gab1∗:P lc∗γ
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42 Kdr∗:Gab1∗:P lc∗γ :P ip2 → Kdr∗:Gab1∗:P lcγ + Dag —————
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45 Kdr∗:Gab1∗:P ip3:P lcγ → Kdr∗:Gab1∗:P ip3:P lc∗γ
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48 Kdr∗:Gab1∗:P ip3:P lc∗γ :P ip2 → Kdr∗:Gab1∗:P ip3:P lcγ + Dag —————
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Gab1∗:P ip3 : P lcγ + Kdr∗ Kdr∗:Gab1∗:P ip3:P lc∗γ k52
Fig. 2. Reactions of the detailed model
In the model Akt and DAG have been considered as the end points of the survival and proliferation pathways, respectively. Taking into account these end points in combination with the notion that Akt activation is strictly P ip3 dependent, we examined the signal
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transduction cascade focusing our attention mainly on the reactions that lead to the production of P ip3 (i.e. k21 and k27 ) and DAG (i.e. k36 , k42 , and k48 ). This analysis (supported also by a careful drawing of the SPN) allowed us to recognize different subpathways that lead to the survival or proliferation effects. In the context of the survival signal module we identified two subpathways, each one characterized by the presence of a distinguishing complex, KDR∗:Gab1∗ or KDR∗:Gab1∗: P ip3 , belonging to the First Block. Actually, the subpathways that determine the survival behavior are three since an additional element, Gab1∗:P ip3 , also contributes to the formation of KDR∗ :Gab1∗ :P i3k:P ip3 complex already involved in one of the identified subpathways. Summarizing there are three subpathways that lead to survival effect starting from: KDR∗ :Gab1∗ , KDR∗ :Gab1∗ :P ip3 and Gab1∗ :P ip3 . Considering the proliferation module, we identified four different subpathways that are distinguished by the compounds belonging to the First Block, i.e.: KDR∗ , KDR∗ :Gab1∗ , KDR∗ :Gab1∗ :P ip3 or Gab1∗ :P ip3 . Notice that the distinguishing elements of the detected subpathways are the same within the survival and proliferation modules, with the exception of the compound KDR∗ . Referring again to the SPN of Fig. 3, we can notice that the time evolution of this SPN is intuitively portrayed by a topdown view. On the top is depicted the place KDR∗ that represents the starting point of the signal cascade induced by its ligand. From the KDR∗ cascade start all the subpathways that characterize the proliferation and survival pathways. The places describing DAG and P ip3 are aligned on the bottom of the net. It is interesting to note that the subpathways we identified in the detailed model are represented in the SPN with structures, such as that outlined by a dashed box in Fig. 3, which correspond to the reaction groups separated by continuous lines in Fig. 2. We denote these SubComponents by SC. Each SC involves: – the binding between an enzyme and other species present in the cascade (e.g. transitions k31 k32 ); – the enzyme activation (e.g. transition k33 ); – the recruitment of the P ip2 (e.g. transitions k34 k35 ); – the production of the molecules representing the pathway end point and the enzyme deactivation (e.g. transition k36 ). 3.3 Model Simplification The SPN we obtained requires a simplification process to take place in order to limit the complexity of the parameterization and analysis of the model as we pointed out before. The computation of the Psemiflows of this SPN show that the net is bounded (the net is covered by Psemiflows, i.e., every place of the net is member of the support of one Psemiflow, at least). Interpreting the Psemiflows in biological terms, we can recall again that this means that all the compounds associated with the places of the net, independently of their original amounts, cannot grow indefinitely during the evolution of the model out of its initial state The presence of repeated structures in the SPN corresponds to the fact that the biological model is characterized by the existence of several similar reaction groups, and this observation can be used to identify simplification steps to be applied to the detailed
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model. In particular, the SCs shown in the previous section are enzymatic kinetics reaction groups. Each of these groups can be written as the reactions set (1) where the enzyme E binds reversibly to the compound C. E + C EC → E ∗ C ∗
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Fig. 4. Reactions after first step of simplification
By exploiting this representation, we rewrite the reactions of the Third and the Fourth Blocks as shown in Fig. 4, and we use them to simplify the original SPN obtaining the net depicted in Fig. 5, that is still covered by Psemiflows, meaning that these transformations are acceptable also from a qualitative point of view. Note that in this new
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SPN any SC is represented by a sequence of transitionplacetransition, such as the one outlined by the dashed box, again. These SCs are defined by reaction groups separated by continuous line in Fig. 4. A further simplification step can be performed on the basis of the following reaction: E + S + S → E + S + P (3) The application of this observation to reactions of the Third and Fourth Blocks provides the reactions illustrated in Fig. 6, used to define the new simplified net illustrated in Fig. 7 (which is again covered by Psemiflows). Note that in this last SPN any SC is represented by a singular transition such as the one outlined again by the dashed box. Moreover reactions k29 k30 , k31 and k37 k38 , k39 of the Second Block are simplified following the scheme: E + S → E + P . The structural criteria that we used to guide the simplification process helped us to verify that the reduced models maintain the biological significance of the original one and provide a good approximation of its behavior. Notice that the reaction substitutions represented by Eq. (2 and 3) can be seen as patterns (or net substructures) that can be replaced every time they occur in the detailed and intermediate nets of the simplification process. In reality, it has not been possible to perform these transformations ”mechanically”. Instead, the invariant conditions had to be checked for each simplification substitution. At the end of this process, considering that the Psemiflows of the simplified SPNs have smaller supports, we found the same eight Psemiflows in all the three nets: – one Psemiflow including KDR∗ and the complexes containing it that are present in the subpathways (both in proliferation and in survival) that bring to the recruiting of substrate P ip2 ;
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– one Psemiflow including Gab1 and the complexes containing it that are present in the subpathways (both in proliferation and in survival) that bring to the recruiting of substrate P ip2 ; – one Psemiflow including Akt and the complexes that lead to its activation; – one Psemiflow including both P ip3 and Dag, and P ip2 that is the common substrate in both pathways. This semiflow includes also the cascade complexes containing P ip3 ; – each enzyme present in the model (P i3k, P lcγ, P ten, E) has a semiflow including the complexes containing it. The consistency among the structural properties of all the nets allowed us to consider the simplified models valid from a qualitative point of view. 3.4 Model Analysis for Accuracy Assessment The simplification process proposed in Section 3.3 results in SPNs which maintain the qualitative properties of the original SPN, but are approximations of the detailed model from a quantitative point of view. In this section we report in silico experiments that were performed in order to check the validity of the simplifications from the point of view of quantitative properties. Indeed, before using the simplified models in a parameter identification experiment which uses real data coming from wetlab experiments, it is necessary to make sure that an overall agreement exists between the quantitative temporal behaviours of the detailed and the simplified models for a wide range of model parameters. This test allows to build confidence on the fact that the reduced model is
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suited for a preliminary analysis of the angiogenesis process as it can be done with in silico experiments. This accuracy assessment was performed applying the “deterministic approach” described in Section 2.3. We compared the temporal behaviours of the detailed SPN with those of the simplified SPNs obtained for several different initial markings and several sets of transition rate parameters. Throughout the comparisons we concentrated on three important entities, in particular P ip3 and Dag, and the substrate P ip2 which is common to both the survival and the proliferation pathways. Hereafter we report on two cases which illustrate the obtained results. The initial condition is identical for the two cases: for all three SPNs (see Figures 3, 5 and 7) we use the initial marking n1 = 2, n2 = n3 = n4 = 1, n5 = 20, n6 = n7 = n8 = n9 = 1 which reflects the concentration differences that are likely to exist in wetlab experiments. Different sets of transition rates are used in the two cases to push the behaviours of the models in opposite directions. In the first case the rates are such that the transitions along the survival pathway are ten times faster than all the others. Figure 8A depicts the temporal behaviour for the detailed model. With these parameters the concentration of P ip3 increases, the concentration of P ip2 decreases and the concentration of DAG remains low. The temporal behaviour of the simplified models, depicted in Figures 8BC, shows the same major characteristics. In the second case the transitions along the proliferation pathway are ten times faster than all the others. Figures 8EFG depict the temporal behaviour for the three models. Also in this case, the major characteristic, i.e. the fact that the concentration of DAG prevails over the concentration of P ip2 and P ip3 , is maintained. The general agreement among the results of all these models was also tested by
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computing the steady state distribution of tokens in the places corresponding to compounds P ip3 and DAG and to the substrate P ip2 , obtained from the solutions of the CTMCs corresponding to all the SPN models we have constructed (that we do not report in detail in this paper, due to space constraints). For the first set of experiments reported here, the rates of corresponding transitions in the simplified and detailed models were set equal independently of the fact that in the simplified SPNs they often represent the compound effect of a few transitions of the detailed one. As a result, even if the overall characteristics are maintained, the shape of the curves can be rather different and the dynamics take place on different time scales. Focusing our attention on the curves corresponding to the first set of parameters (Figures 8ABC), we can notice that the crossing of the two concentration curves for P ip2 , which decreases, and P ip3 that grows takes place at time instants that are not of the same order of magnitude in all the three cases. A more important difference, however, is observed when we concentrate on the dynamics of DAG that shows a small initial growth followed by a descent to a value next to zero. For this case, the most simplified model predicts an important initial climb that makes the shape of the curve quite different from the others (see Figure 8C). Turning our attention to the curves corresponding to the second set of parameters (Figures 8EFG), we can notice that all the models predict a crossing between the concentration of P ip2 which decreases and the
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concentration of DAG that instead grows. With these parameters the concentration of P ip3 remains always extremely small. In this case the predictions of the more compact model are quantitatively quite different since the crossing is reported to happen much sooner and the shapes of the curves are very different. In order to mimic better the temporal behaviour of the detailed SPN, we applied an optimization technique to determine the transition rates for the simplified SPNs. We used the nonlinear optimization function of MatLab to find these transition rates with which the curves resulting from the simplified SPNs became closer to the curves resulting from the detailed SPN. The results are illustrated for the SPN obtained after the second step of simplification (Simpl2) in Figure 8D and 8H. It can be seen that by properly setting the transition rates, even the simplest of our SPNs (Simpl2) can mimic quite precisely the behaviour of the detailed model (compare the diagrams A with D and E with H of Figure 8).
4 Discussion In this paper we showed how to use structural properties of SPNs and biochemical properties of the system in guiding the simplification process. The procedure was presented through a case study, namely, the model of signaling transduction pathways involved in the angiogenesis process. We showed that the procedure results in simplified SPNs that are able to mimic precisely the temporal behavior of the detailed SPN. One non trivial step is to determine the transition rates in the simplified SPNs in such a way that the resulting temporal behavior is a good approximation of that of the detailed SPN. In this work we faced this problem by applying optimization. In the future, on the basis of the simplification schemes presented in this paper, we plan to work on generalizations of these reduction steps which will allow to operate on other portions of the detailed model and on the identification of rules concerning the relations existing among the corresponding rates of the detailed and simplified models. In addition, we will study the possibility of defining formally the quantitative characteristics that have to be maintained by the simplification process. In particular, temporal logics will be considered to this purpose. Furthermore, we will consider the study of the whole VEGFinduced intracellular network, including signal modules that were not considered here, such as the migration pathway. This could contribute to a better understanding of the intricate signaling induced by VEGFA during the angiogenesis.
Acknowledgment Cordero is the recipient of research fellowship supported by grants from Italian Association for Cancer Research and Regione Piemonte.
References 1. Baarir, S., Beccuti, M., Cerotti, D., De Pierro, M., Donatelli, S., Franceschinis, G.: The GreatSPN tool: recent enhancements. SIGMETRICS Perform. Eval. Rev. 36(4), 4–9 (2009) 2. Balbo, G.: Introduction to stochastic petri nets. In: Brinksma, E., Hermanns, H., Katoen, J.P. (eds.) EEF School 2000 and FMPA 2000. LNCS, vol. 2090, p. 84. Springer, Heidelberg (2001)
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3. Busch, H., Sandmann, W., Wolf, V.: A numerical aggregation algorithm for the enzymecatalyzed substrate conversion. In: Priami, C. (ed.) CMSB 2006. LNCS (LNBI), vol. 4210, pp. 298–311. Springer, Heidelberg (2006) 4. Calder, M., Vyshemirsky, V., Gilbert, D., Orton, R.: Analysis of Signalling Pathways Using Continuous Time Markov Chains. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 44–67. Springer, Heidelberg (2006) 5. Dance, M., Montagner, A., Yart, A., Masri, B., Audigier, Y., Perret, B., Salles, J.P., Raynal, P.: The adaptor protein Gab1 couples the stimulation of vascular endothelial growth factor receptor2 to the activation of phosphoinositide 3kinase. Journal of Biological Chemestry 281, 23285–23295 (2006) 6. Gerber, H.P., McMurtrey, A., Kowalski, J., Yan, M., Keyt, B.A., Dixit, V., Ferrara, N.: Vascular endothelial growth factor regulates endothelial cell survival through the phosphatidylinositol 3’kinase/Akt signal transduction pathway. requirement for Flk1/KDR activation. Journal of Biological Chemestry 273, 30336–30343 (1998) 7. Gillespie, D.T.: A rigorous derivation of the master chemical equation. Physica 188, 404–425 (1992) 8. Glbert, D., Heiner, M., Lehrack, S.: A unifying framework for modelling and analysing biochemical pathways using Petri nets. In: Proc. Int. Conf. Computational Methods in System Biology, pp. 200–216 (2007) 9. Goss, P., Pecoud, J.: Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets. Proc. Natl. Acad. Sci. 95(12), 6750–6755 (1998) 10. Heiner, M., Koch, I., Will, J.: Model validation of biological pathways using Petri nets demonstrated for apoptosis. BioSystems 75, 10–28 (2004) 11. Hofest¨adt, R.: A Petri net application of metabolic processes. Journal of System Analysis, Modeling and Simulation 16, 113–122 (1994) 12. Hofest¨adt, R., Thelen, S.: Quantitative modeling of biochemical networks. Silico Biology 1(6) (1998) 13. Laram´ee, M., Chabot, C., Cloutier, M., Stenne, R., HolgadoMadruga, M., Wong, A.J., Royal, I.: The scaffolding adapter Gab1 mediates Vascular Endothelial Growth factor signaling and is required for endothelial cell migration and capillary formation. Journal of Biological Chemistry 282, 7758–7769 (2007) 14. Memmi, G., Vautherin, J.: Advanced algebraic techniques. In: Brawer, W., Reisig, W., Rozenberg, G. (eds.) APN 1986, Part I. LNCS, vol. 254. Springer, Heidelberg (1987) 15. Molloy, M.K.: On the integration of delay and throughput measures in distributed processing models. Ph.D. Thesis, UCLA (1981) 16. Natkin, S.: Les r´eseaux de Petri stochastiques et leur application a` l’´evaluation des syst`emes informatiques. Th`ese de Docteur Ing´egneur, CNAM (1980) 17. Reddy, V., Mavrovouniotis, M., Liebman, M.: Petri net representation in metabolic pathways. In: Proc. Int. Conf. Intelligent Systems for Molecular Biology, pp. 328–336 (1993) 18. Reisig, W.: A Primer in Petri Net Design. Springer Compass International, Heidelberg (1992) 19. Takahashi, T., Ueno, H., Shibuya, M.: VEGF activates protein kinase Cdependent, but Rasindependent RafMEKMAP kinase pathway for DNA synthesis in primary endothelial cells. Oncogene 18, 2221–2230 (1999)
CSL Model Checking of Biochemical Networks with Interval Decision Diagrams Martin Schwarick and Monika Heiner Department of Computer Science, Brandenburg University of Technology Postbox 10 13 44, 03013 Cottbus, Germany
[email protected],
[email protected] Abstract. This paper presents an Interval Decision Diagram based approach to symbolic CSL model checking of Continuous Time Markov Chains which are derived from stochastic Petri nets. Matrixvector and vectormatrix multiplication are the major tasks of exact analysis. We introduce a simple, but powerful algorithm which uses explicitly the Petri net structure and allows for parallelisation. We present results demonstrating the eﬃciency of our ﬁrst prototype implementation when applied to biochemical network models, speciﬁcally with increasing token numbers. Our tool currently supports CSL model checking of timebounded operators and the Next operator for ordinary stochastic Petri nets.
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Stochastic Petri nets are a natural way to model biochemical networks, where token values may be interpreted as molecules or concentration levels [GHL07], [HGD08]. Petri nets reﬂect explicitly the network structure, which contributes to a better understanding of the network behaviour, and – as we are going to see – supports eﬃciency gains otherwise not possible. A stochastic Petri net’s semantics is a Continuous Time Markov Chain (CTMC) which can be investigated by simulative approaches, or analysed analytically by transient and steadystate analysis [Ste94], or model checking of Continuoustime Stochastic Logic (CSL) [ASSB00]. In this paper we concentrate on (analytic) CSL model checking, which has been proven to be particularly useful for model validation and modelbased experiment design in systems and synthetic biology: special behavioural properties are expressed in CSL, a ﬂexible and powerful query language, and then checked exhaustively against all behaviour the model can exhibit. The tool of choice when applying CSL model checking of CTMCs is often the probabilistic model checker PRISM [PNK06], which seems to represent the current state of the art [JKO+ 08]. Stochastic Petri nets can be easily translated into the PRISM input language as it has been done in [CDDS06], [GHL07], [HGD08]. However, computational experiments reach pretty fast their limits, as they always do if the famous state space explosion problem is one of the game players. P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 296–312, 2009. c SpringerVerlag Berlin Heidelberg 2009
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PRISM’s approach to cope with the problem is symbolic analysis based on Multi Terminal Binary Decision Diagrams (MTBDD), which are basically Binary Decision Diagrams (BDD) allowing more than two terminal nodes, each standing for a diﬀerent value. While this often works ﬁne for technical systems resulting into 1bounded networks, it does not smoothly scale to the generalised bounded case. First of all, prior knowledge of the boundedness degree of each place is required. A place with an upper bound of k tokens is represented by ld(k) MTBDD variables. This may result in an overhead in computation time and memory. Since tokens may represent concentration levels, increasing the analysis accuracy implies an increase of the possible number of tokens on places. Secondly, PRISM creates an MTBDD which represents the entire CTMC with states and transitions encoded in a matrix. Therefore it is necessary to double the number of MTBDD variables to index rows and columns. Finally, a further drawback occurs if the CTMC contains many diﬀerent rate values, since the number of terminal nodes in the MTBDD equals this amount. These lessons learnt from the PRISM approach made us elaborate a new technique for symbolic CSL model checking, speciﬁcally designed for biochemical networks with increasing token numbers. The eﬃcient analysis of qualitative Petri nets, provided they are bounded, but not necessarily 1bounded, is discussed by A. Tovchigrechko in [Tov08]. He deploys Interval Decision Diagrams (IDD), which generalise BDDs by allowing more than two outgoing arcs for each node, but keeping the idea of two terminal nodes only. The developed data structures and algorithms support state space based analysis, including model checking of Computational Tree Logic (CTL). They do neither require a priori knowledge of the boundedness degree nor a suitable network partitioning as Kroneckerbased approaches do, see e.g. [CJMS06]. The IDDs’ inherent compression eﬀect often yields compact representations of very large state spaces [HST09], see also caption of Table 2 in Section 4. In this paper we are going to demonstrate how these IDD techniques can be transfered and adapted to CSL model checking, which basically requires to incorporate matrixvector multiplication. In doing so we always bear in mind the option of parallelised processing on nowadays standard workstations. It goes without saying, the application of our results is not restricted to stochastic Petri nets. Speciﬁcally we will demonstrate how PRISM’s eﬃciency may take advantage of our preanalysis of a network’s inherent structure.
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Stochastic Petri Net. An ordinary stochastic Petri net SP N is a tuple (P, T, F, V, s0 ). As usual, P denotes the set of places, T the set of transitions, F : ((P × T ) ∪ (T × P )) → {0, 1} the arc weight function, and s0 the initial state (marking). The mapping V : T → H, where H is the set of hazard functions, associates to each transition a function ht from H, deﬁning a generally statedependent, but always exponentially distributed ﬁring rate. We deal with biologically interpreted stochastic Petri nets; thus we consider besides arbitrary arithmetic functions speciﬁcally functions representing biomass action semantics (BMA) and biolevel interpretation semantics (BLI). All these functions have in
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common that the domain is restricted to the preplaces of the corresponding transition. For more details see [GHL07]. Continuous Time Markov Chain. The semantics of a stochastic Petri net is a CTMC which is isomorphic to the reachability graph of the underlying qualitative Petri net, but state transitions are labelled with ﬁring rates. Without t
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The total rate E(s) = Σs ∈S R(s, s ) is the sum of entries of the matrix row indexed with s. A state s with E(s) = 0 is called an absorbing state, since there is no way to leave it when reached. The probability of a transition t enabled in state s to ﬁre (which results in state s ) within n time units is 1 − e−R(s,s )·n . The transient probability π(α, s, τ ) is the probability to be in state s at time τ starting from a certain probability distribution α, with α : S → [0, 1] and Σs∈S α(s) = 1. The vector of transient probabilities for all states at time τ with the initial distribution α is denoted by π(α, τ ). An established technique to compute the transient probabilities (transient analysis) of CTMCs is the uniformisation method. Its basic operation is vectormatrix multiplication which must be done for a certain number of iterations. For more details see [Ste94]. Continuous time Stochastic Logic. CSL is the stochastic counterpart to Computation Tree Logic (CTL). We consider CSL without the steady state operator and timeunbounded path formulae, and deﬁne state formulae φ ::= a  ¬φ  φ ∧ φ  φ ∨ φ  Pp [ϕ] , and path formulae ϕ ::= Xφ  φ U[τ1 ,τ2 ] φ  F[τ1 ,τ2 ] φ  G[τ1 ,τ2 ] φ , with a ∈ AP , ∈ {}, p ∈ [0, 1], and τ1 , τ2 ∈ R≥0 ∧ τ1 ≤ τ2 ∧ τ2 < ∞. For convenience we introduce the operators F φ and G φ as shorthand notations for the frequently used patterns true U φ, and ¬(true U ¬φ). CSL model checking of a CTMC M can be realised by transient analysis. The basic concept is to do transient analysis for a CTMC M which has been derived from M by making certain states absorbing, depending on the formula to be checked. For more details, e.g. formal semantics deﬁnition, see [BHHK00].
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Interval Decision Diagrams. An IDD is a rooted, directed and acyclic graph with nodes having an arbitrary number of outgoing edges. Each edge is labelled with a leftclosed and rightopen interval on N0 . The intervals of the outgoing edges of each IDD node deﬁne a partition of N0 , inducing a total order of the edges. There are two nodes without outgoing edges: the terminal nodes, labelled with ONE and ZERO. Each IDD node gets associated a variable, in our context a place of the stochastic Petri net. We assume that the variables occur in the same order on each path from the root to a terminal node – we get ordered IDDs. Furthermore we assume that an IDD does not contain isomorphic subgraphs – we get reduced ordered IDDs. As for BDDs, the variable ordering may inﬂuence the IDD size. We use IDDs to encode sets of states of stochastic Petri nets, see Figure 1. The height of an IDD always equals the number of places, independently of the places’ boundedness degree. IDD’s grow in the breadth: a large variety of tokens on a given place may increase the number of outgoing edges of the corresponding IDD nodes, depending on the IDDinherent compression eﬀect. We consider bounded Petri nets; thus, each IDD node for a kbounded place has at least two outgoing edges: [0, k+1), and [k+1, ∞). A path (sequence of IDD nodes connected by edges) reaching the terminal node ONE represents generally a set of states. We get one state by choosing exactly one value from each of the intervals of all edges occurring along a path. For the eﬃcient manipulation of state sets we assume operations like ∩, ∪, \. Further we assume operations for the manipulation of state sets by the ﬁring t of transitions. F ire(S, t) := {s s ∈ S ∧ s − → s } represents the set of states n7(p1)
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obtained by ﬁring the transition t for each state in S, and Img(S) represents the set of all direct successor states of S. Analogously, we deﬁne RevF ire(S, t) and P reImg(S) for backward ﬁring. For more details see [Tov08]. While CTL model checking can be completely reduced to the manipulation of sets of integer states, CSL model checking by transient analysis requires the (repeated) multiplication of a realvalued matrix with a realvalued vector (or vice versa). On account of the state space explosion problem it is not worth thinking about implementing this vectormatrix multiplication explicitly with matrix and vector indexed by states. There is no way to avoid the explicit representation of the vector π. Actually, we need at least three (four) copies of it. Thus, the whole problem boils down to the question: How to multiply with a matrix without having (explicitly represented) the matrix? In the following we present a matrixfree ontheﬂy approach to realise CSL model checking more eﬃciently than other tools available so far. Our tool IDDCSL computes all required data at each iteration anew from one augmented IDD representing the reachable states of an SPN. Thus, our technique does not care about the number of diﬀerent matrix entries in the rate matrix. This is – in terms of data structures – the main diﬀerence to PRISM’s approach, where the CTMC’s state space S and its rate matrix R are represented symbolically by a BDD and an MTBDD.
3 3.1
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We do not use dedicated data structures to represent the CTMC as other symbolic model checkers do. All necessary information is derived from the set of reachable states encoded as an IDD and the Petri net structure itself. However, we do need the lexicographic index for each state in the state set, which will be determined by each depth ﬁrst search traversal of the decision diagrams. One slight extension of the IDD is required to get these indices, which brings us to the indexlabelled IDD, LIDD for short. Now the basic idea of our approach is simply explained. The traversal of an IDD representing a state set S ⊆ S drives the traversal of the LIDD, representing S. This indirect, partial traversal of the LIDD S allows to compute the index for each state s ∈ S . Additionally we keep track of the index of the state s , reached by ﬁring a given transition t ∈ T in s , assuming s enables t. We compute the enabling states for a transition t by ESt := S∩RevF ire(S, t). When traversing an IDD, we always consider the pre and postconditions for the ﬁring of a transition t of the underlying Petri net to determine the LIDD paths of the related target states. This idea is inspired by the ﬁre algorithm proposed in [Tov08]. Each traversal extracts the indices of all state transitions (matrix entries) of the CTMC induced by the ﬁring of a transition t. Traversing the LIDD for all transitions controlled by their enabling states eventually extracts all nonzero matrix entries. Thus, each iteration required for the transient analysis means the LIDD traversal for all transitions of the Petri net.
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Algorithm 1. LIDD index labelling procedure augmentIdd(states : LIdd) augmentNode(states.root); end procedure function augmentNode(node : LIddN ode) int : if node = ON E then return 1; end if if node = ZERO then return 0; end if count, reachable : int; count := 0; for 0 ≤ i < node.edges() do edge : LIddEdge; edge := node.edge(i); edge.smaller = count; reachable :=augmentNode(edge.node()); count := count + reachable ∗ edge.intervalW idth(); end for return count; end function
Augmentation of an IDD with index information. The required state indexing calls for an augmentation of the IDD, representing the reachable states, by some information which allows the necessary computation. Inspired by [MC99] we store the amount of lexicographic smaller states for each edge, which can be reached by all its previous sibling edges. This style to organise the index information allows to keep several sets within one and the same LIDD with diﬀerent sets having diﬀerent nodes as root. Algorithm 1 sketches how to derive recursively the LIDD representing S. The generated additional index data are labelled with a pound (#) in Figure 1. Determining a matrix entry. We need the value (rate) of the current matrix entry R(i, j) to multiply the rate matrix with a vector or vice versa. This rate is determined by the hazard function of the Petri net transition which is responsible for the state transition from the state with index i to the state with index j. Consequently, while computing the index pair for each state transition, we have to compute this function value, too. See Algorithm 2. Manipulating the matrix. Please recall, model checking of timebounded CSL operators for a CTMC M can be reduced to the problem of applying transient analysis to M which has been derived from M by making certain states absorbing. For this purpose, PRISM creates a new MTBDD representing the rate matrix of M . In our approach this means only to call the procedure traverse for the nonabsorbing subset N ESt of the enabling states ESt of a transition t. When A is an IDD representing the set of absorbing states, N ESt can be computed eﬃciently by ESt \ A. Model checking the X operator involves the socalled Embedded Markov Chain (EMC). The EMC is a Discretetime Markov Chain (DTMC), i.e. transitions are labelled with probabilities and the rate matrix R is replaced by the
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Algorithm 2. LIDD traversal procedure traverseAllTransitions //the following for loop can be parallelised t : T ransition; ESt : Idd; for 0 0 then smaller := node.edge(edgeIndex − 1).smaller; end if return smaller + (val − edge.lowerBound()) ∗ edge.node().lastEdge().smaller; end function
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probability matrix P, where each entry (s, s ) represents the probability of a state transition from s to s . The sum of each row of P is 1. The EMC Me is derived from a CTMC M by deﬁning P as P(s, s ) := R(s, s )/E(s). We assume that the values E(s) for all states are stored in a vector. Multiplying the EMC with a vector means to adapt Algorithm 2 by the code given in Algorithm 3. This approach should also work for PRISM’s traversal algorithm. PRISM uses a further MTBDD to represent the EMC, for which the number of terminal nodes and thus the overall number of nodes can explode (see Section 4). Algorithm 3. Adaptation of Algorithm 2 to handle Embedded Markov Chains if root = ONE then // e.g. vectormatrix r=v*M : // r[srcIndex] = v[destIndex]*rf.compute(fa) // rate is M[srcIndex][destIndex] rateEmbedded : double; rateEmbedded := t.rateF unction.compute(f a)/E[srcIndex]; processData(srcIndex, destIndex, rateEmbedded); return ; end if
3.2
Optimization Techniques
In this section we sketch some optimization techniques contributing to the eﬃciency of our approach. Variable Ordering. It is well known that the chosen variable order is crucial for the size of decision diagrams and thus for the eﬃciency of related algorithms. [Noa99] suggests a greedy algorithm to obtain a static variable order for Zero Suppressed Binary Decision Diagrams which is based on heuristics exploiting the Petri net structure. The basic idea is to create an order where related variables are close together. Related variables are in our case places, which are directly connected by a Petri net transition. The heuristic algorithm creates stepwise an order ω, starting at the lowest IDD level and using the weight function W (p) to determine the next place from the set of unprocessed places to be inserted in ω based on the set of already processed places Q. Using the standard dot notation to specify the set of pre or postnodes of a given node we deﬁne W (p) by: W (p) :=
Σt∈• p 
•
t∩Q t• ∩Q • • t + Σt∈p t•   •p ∪ p • 
.
(1)
Our approach beneﬁts from the observation that the variable orders obtained by this algorithm usually yield small IDDs, see [Tov08], [HST09], and Section 4. A prominent heuristics to represent matrices as MTBDDs relies on variable orders with alternating row and column variables. Additionally, it is worthwhile to ﬁnd a good overall variable order, as we will see in Section 4. PRISM reads models as they are, i.e. it will not change the order of modules and the variables therein contained. Using the sketched ordering algorithm when specifying
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PRISM models generally speeds up the state space construction and the model checking signiﬁcantly. Caching. As for every implementation of decision diagrams, eﬃciency depends on considering redundancies. Generally, nodes on lower IDD levels will be visited many times. Subpaths beginning in these nodes will be traversed each time anew. Following [Par02] we set a certain layer of the LIDD and cache index and rate information for each of its nodes for all paths to reach this node. Each time a node of the cache layer is reached, the cache data must be retrieved only. Our algorithm traverses the LIDD transitionwise driven by an unlabelled IDD representing the enabling states of the transition or a nonabsorbing subset of it. Thus it is necessary to store transitionspeciﬁc information for all LIDD nodes of the cache layer. The cache data contain for each transition the index pair and the rate function of all possible path extensions. Each visit of a cache layer node comes with a unique index pair and a unique set of function arguments which allow to compute all related matrix entries using the cache data. A cache datum consists of an index pair and a rate function. Often many index pairs refer to the same function. Thus we associate to a function a set of index pairs. In general, a cache layer node keeps several rate function instances and their assigned index pairs for each transition. Changing the enabling states or the rates of the CTMC, e.g. by uniformisation, needs to reinitialise all cache data. To use cache data requires a modiﬁcation of Algorithm 2. When visiting a cache layer node, the transitionrelated cache data will be processed and the procedure traverse returns, compare Algorithm 4. A crucial point for an implementation of this approach is to store the cache data, in particular the index sets, as memorysaving as possible. A naive way of doing this is to store lists of index pairs. But a closer look to the possible values reveals that there are often consecutive pairs with a ﬁxed step size. This is a consequence of the fact that we obtain a huge state space by ﬁlling a Petri net with tokens, but without changing its structure. If such sequences of consecutive index pairs exceed a critical length it is worthwhile to represent them by a tuple (f irst rowIndex, f irst colIndex, row stepSize, col stepSize, steps). Then, the sequence (0, 0); (5, 10); (10, 20); . . . ; (100, 200) can be encoded by the tuple (0, 0, 5, 10, 20). An issue here is to ﬁnd a suitable critical length. Traversal for transition sets and arbitrary state sets. The basic algorithm sketched so far requires a separate traversal for each transition of the stochastic Petri net. An improvement is to generalise the algorithm such that it controls the traversal of the LIDD S for a set of transitions and an IDD encoding an arbitrary set of states S ⊆ S. Then the algorithm must treat lists of source and target indices. The lists contain for each transition an entry holding the current traversal data. The basic algorithm ensures that the traversalcontrolling IDD contains enabling states of a transition only. A generalization of the algorithm must deal with disabling states, too. Our prototype tool IDDCSL implements the generalised algorithm.
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Algorithm 4. Adaptation of Algorithm 2 in order to use cache data if cacheLayerReached() then rate : double; rf : RateF unction; indices : IndexSet; actSrcIndex, actDestIndex : int; cd : CacheData; cd := src.cacheData(t); for 0 ≤ i < cd.entries() do rf := cd.getF unction(i); indices := cd.getIndices(i); rate := rf.compute(f a); for 0 ≤ j < indices.size() do actSrcIndex := srcIndex + indices.getSrcIndex(j); actDestIndex := destIndex + indices.getDestIndex(j); processData(actSrcIndex, actDestIndex, rate); end for end for return ; end if
Parallelisation. Today’s workstations or even standard personal computers in an everyday secretary oﬃce tend to possess two or more processors. Thus it is appealing to take advantage of the available multiple processors. There are basically two approaches to divide the problem of a matrixvector multiplication or vice versa into smaller tasks, which can be solved concurrently. On the one hand one could divide the Petri net’s transition set and apply the algorithm concurrently to each subset. Doing so obviously requires some kind of synchronisation techniques. On the other hand one could partition the state space. Applying our algorithm concurrently with forward (backward) ﬁring transitions with a partitioned state space means to devide the matrix rowwise (columnwise) into submatrices and requires no synchronisation when doing a matrixvector (vectormatrix) multiplication, because each row (column) is considered for all transition by only one thread. When synchronisation is required all threads get their own complete result vector and collect the results of all other threads after each computation phase. Although parallelisation is not the focus of this paper, we are going to indicate its potential by presenting some related results in the following section.
4
Benchmarks
In this section we present results comparing our prototype implementation IDDCSL with PRISM, and by transitivity with a couple of CSL model checking tools on the market [JKO+ 08]. As benchmarks we consider stochastic Petri nets of the following popular biochemical networks. – The mitogenactivated protein kinase (MAPK) cascade published in [LBS00] and discussed as three related Petri net models in [GHL07], [HGD08]. All initial states considered in our paper are multiples of level 4. This is the minimal initial (integer) state respecting the ratio in the initial (realvalued)
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concentrations as given in [LBS00]. Our model is structurally identical with the MAPK cascade given on the PRISM website. The models only diﬀer in the names of variables, the initial state and the speciﬁed rate constants. – The RKIP inhibited ERK pathway (ERK) published in [CSK+ 03], analysed with PRISM in [CVOG05], discussed as qualitative and continuous Petri nets in [GH06], and as three related Petri net models in [HDG10]. – The circadian clock model (CC) published in [BL00] and available as PRISM model on the PRISM website. For the comparison with PRISM we either use the export feature of our modelling tool Snoopy [Sno08] (MAPK, ERK) or an available PRISM model (CC). The latter example needs capacities to enforce boundedness, which we simulate in the Petri nets by complementary places. All models have scalable initial states. The experiments consider biomass action (resp. biolevel interpretation) semantics, for which IDDCSL oﬀers the predeﬁned BioM assAction (resp. BioLevelInterpretation) function. Our implementation makes use of Intel’s instruction set extension SSE2 which could also speed up PRISM. In some cases the eﬃciency gain is about 10 percent. Our test system is a Dell Precision workstation with 4 GB main memory and an Intel Xeon with 4 × 2.83GHz running a 64bit Linux. In our computational experiments we focus on runtime. All related ﬁgures are given in seconds. The influence of variable order. In contrast to the modelling style in [KNP08], our generated monolithic PRISM models consist of one module only, with a module variable for each place. The value range of the variables (boundedness degree) and the variable order were computed by our IDDbased tool box. Table 1 illustrates the impact of the chosen variable order on PRISM’s eﬃciency for diﬀerent levels of the MAPK cascade, and Table 2 the CTMC size for diﬀerent levels computed with PRISM using a good variable order. Table 1. Comparison of two variable orders. The table shows the time and the number of MTBDD nodes, which PRISM needs to construct the rate matrix of the CTMC for a good variable order,computed using formula (1), and for the plain order of the original PRISM model, speciﬁed according to [KNP08]. levels
terminal good order original order nodesa) time nodes time nodes 4 30 0.12 8,672 2.47 123,730 8 76 1.56 60,452 401.68 3,881,914 12 140 22.99 199,496 16 219 71.25 542,339 20 320 296.87 953,146 24 453 635.92 2,029,598 28 697 928.45 3,771,617 32 770 1847.60 6,015,521 a) i.e., number of diﬀerent entries in rate matrix; ’’ exceeds the available memory;
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Table 2. CTMC size for diﬀerent levels computed with PRISM using a good variable order, see Section 3.2. In principle our tool IDDCTL allows to compute the state space up to level 320 (2.627e+27 states ) in about 2 minutes on a standard personal computer [HST09]. However, the transient analysis it limited to the available memory to store the required vectors π(α, τ ) in the size of the state space. levels
number of edgesa)
number of states
4 24,065 8 6,110,643 12 315,647,600 16 6,920,337,880 20 88,125,763,956 24 769,371,342,640 28 5,084,605,436,988 32 27,124,071,792,125 i.e., number of nonzero entries in rate matrix;
a)
206,007 78,948,888 4,958,809,056 122,381,517,819 1,689,018,298,500 15,635,976,824,982 108,065,356,604,208 597,236,499,605,178
Table 3. The formula P>0.0 [F[0,1] Raf P = 2] is true for all states for which the probability is not zero to reach a state within one time unit which satisﬁes Raf P = 2. For model checking of this formula all states satisfying Raf P = 2 become absorbing; there are 1, 083, 102 of them. The derived CTMC M comprises 64, 368, 742 state transitions. b)
cl
PRISMa) totalc)
iterd)
cl
IDDCSL 1 thread total iter 440.23 170.06 158.65 110.02 93.84 79.48 84.62 75.51 84.49 75.04 90.65 81.73 100.40 91.39 127.72 118.09 253.60 243.05 692.84 676.05 1808.66 1771.00
2 threads total iter 432.77 157.08 158.71 99.75 72.71 55.01 62.57 50.55 60.81 49.17 64.08 52.18 67.47 55.97 81.40 69.71 147.85 134.09 387.62 368.08 957.26 917.07
3 208.35 140.82 5 222.67 169.76 7 201.23 154.94 9 200.13 158.99 10 195.83 159.90 11 198.43 163.03 13 214.64 179.90 15 226.16 191.22 17 218.51 184.06 19 230.66 195.92 21 2318.86 2275.71 a) using a good variable order, determined by the network structure, see Table 1; b) cache layers; c) includes time for state space construction, initialisation, computation and determining the satisfying states; d) eﬀective probability computation time; 65 60 55 50 45 40 35 30 25 20 1
The influence of caching. Tables 3 and 4 compare the runtime of IDDCSL and PRISM with diﬀerent cache layers1 for the eight level version of the MAPK cascade with biolevel interpretation semanctics. We use the ﬂat PRISM model with a good variable order, compare Table 1, for these experiments. We take formulae which diﬀer only in the speciﬁed time intervals. The interval [0, 1] – in 1
In PRISM the highest cache layer is the root node layer, in IDDCSL it is the terminal node layer. PRISM’s hybrid engine is used.
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Table 4. The formula P>0.0 [F[1,1] Raf P = 2] is true for all states for which the probability is not zero to be at time 1 in a state which satisﬁes Raf P = 2 . Any path formula F[τ,τ ] φ is suitable to trigger transient analysis up to time point τ using CSL model checking. PRISMa) total
IDDCSL 1 thread 2 threads total iter total iter 3 382.00 148.05 365.91 129.08 65 242.80 163.05 5 135.06 94.01 124.50 74.27 60 231.65 170.69 55 275.53 219.89 7 88.74 75.84 67.32 50.43 9 82.22 73.47 60.25 48.34 50 222.23 173.32 10 81.58 72.82 58.55 47.21 45 210.13 167.48 40 209.91 168.83 11 87.67 78.98 62.42 51.11 13 97.53 88.98 65.55 54.34 35 212.23 171.38 15 124.67 116.00 78.85 67.56 30 211.43 170.39 17 260.95 250.09 150.66 137.09 25 221.78 181.28 19 782.22 766.00 428.38 409.02 20 231.90 192.46 21 2128.20 2087.00 1150.41 1106.00 1 2745.47 2691.70 a) using a good variable order, determined by the network structure, see Table 1; cl
iter
cl
contrast to the interval [1, 1] – generally results into a set of absorbing states due to the CSL model checking algorithm [BHHK00]. A high amount of absorbing states reduces memory consumption and run time. The MTBDD needs 66 row and 66 column Boolean variables. The IDD needs – independently of the number of levels – 22 integer variables, i.e. as many as there are places in the Petri net model. Thus, the MTBDD hight is 132, and the IDD hight is 22. The tables show the total processing time and the eﬀective iteration time for the computation of the probability vector π(α, 1), which requires 218 iterations for each experiment of the used formulae. The best results in terms of total time and iteration time, depending on the used cache layer are highlighted in bold. The last line in each table represents the case where caching is disabled. Further results. Table 5 and 6 present results for the transient analysis using CSL model checking for the ERK model and the circadian clock model with biomass action semantics. We give the total run time for both tools. In general PRISM’s explicit sparse matrix engine is faster then its hybrid MTBDD engine at the expense of a higher memory usage [JKO+ 08]. The tables show that our tool outperforms also the sparse engine. The CTMC size aﬀects the model checking performance. Thus, a high amount of absorbing states (which depends on the CSL formula) may signiﬁcantly speed up the model checking. Except from choosing the engine or the number of threads, we run the tools with their default settings. Please note that changing, e.g., the cache layer would aﬀect the run time and the memory consumption. We also performed experiments with the X operator; we report here of one of them, which relates to the MAPK cascade. The formula P0.0 [F[1,1] Raf 1Star = 1] is true for all states for which the probability is not zero to reach a state within one time unit which satisﬁes Raf 1Star = 1. level
CTMC size PRISM states edges hybrid 5 1,974 12,236 0.73 10 47,047 372,372 5.52 15 368,220 3,213,408 † 20 1,696,618 15,609,594 † 25 5,723,991 54,438,930 † 30 15,721,464 152,964,146 † † time for initialisation exceeds 24 hours;
sparse 0.65 3.97 † † † †
IDDCSL 1 thread 2 threads 0.20 0.17 1.27 1.05 20.73 16.67 148.92 118.59 740.28 581.39 3,005.62 2,455.57
Table 6. CSLbased transient analysis of the circadian clock for several initial markings. The formula P>0.0 [F[1,1] a = 1] is true for all states for which the probability is not zero to reach a state within one time unit which satisﬁes a = 1. level
CTMC size PRISM states edges hybrid sparse 5 31,104 290,160 3.90 2.36 10 644,204 6,766,320 122.55 64.94 15 4,194,304 45,972,480 1,090.44 570.43 20 16,336,404 183,032,640 5,569.65 2,835.89 25 47,525,504 539,650,800 † 30 114,516,604 1,312,110,960 † − exceeds the physical memory; † time for initialisation exceeds
IDDCSL 1 thread 2 threads 1.76 1.16 44.65 26.10 466.47 312.83 2,471.70 1684.96 8,595.37 6,027.33 26,085.95 17,314.66 24 hours;
a state which satisﬁes Raf P = 2. To represent the Embedded Markov Chain PRISM creates an MTBDD which comprises 48, 149, 682 nodes, among which are 217, 974 terminal nodes. The model checking takes 201.09 seconds including state space construction and initialization. IDDCSL requires ﬁve seconds. The ﬁgures speak for themselves. The gap between PRISM and IDDCSL gets larger with increasing amount of levels (tokens, boundedness degree). Our data structure is less sensitive to increasing the amount of levels, and does not care about the amount of diﬀerent matrix entries in the rate matrix.
5
Technicalities
The tool is implemented in C++ , reusing our IDDbased CTL model checking implementation IDDCTL [HST09] and the GNU MB Bignum Library (GMP). The parsing of CSL formulae has been generated by the lexical analyser and parser generator f lex and bison. For parallelisation we use the POSIX pthread library. The tool comes as an allinclusive binary (statically linked libraries) for our development and reference test system Linux. Versions for Windows and Mac/OS are in preparation.
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The Petri net models have been constructed using Snoopy [Sno08], [HRS08], a tool to design and animate or simulate hierarchical graphs, among them stochastic Petri nets as used in this paper. Snoopy provides export to various analysis tools, recently complemented by PRISM, as well as import and export of the Systems Biology Markup Language (SBML). The tools are available at wwwdssz.informatik.tucottbus.de, free of charge for scientiﬁc purposes. At the same web site you ﬁnd also the Petri net examples (in Snoopy, APNN, and PRISM syntax), which we used as benchmarks in the preceding section. Thus, all reported computational experiments can be easily repeated.
6
Conclusions
We have presented a new tool for symbolic CSL model checking of ordinary stochastic Petri nets. We combine approved heuristics with an innovative approach to represent symbolically a CTMC’s rate matrix by Interval Decision Diagrams. We accept potentially higher computational costs in favour of smaller data structures. The models have to be bounded, however, no a priori knowledge of the precise boundedness degree is required. Likewise, we do not depend on a suitable partitioning as Kroneckerbased approaches do. A crucial point for the tool’s performance are the algorithms exploiting knowledge of the network structure. The implementation beneﬁts in particular from the chosen static variable order which also increases PRISM’s performance signiﬁcantly. In total we gave the results of more than 100 computational experiments. The presented benchmarks show that our data structure used for the symbolic state space representation is relatively insensitive to increasing token numbers. The IDD hight is completely deﬁned by the number of variables. The IDD breadth may increase with increasing token numbers, but this depends on the IDD compression eﬀect. Our approach is not sensitive at all to an increasing amount of diﬀerent entries in the rate matrix. In summary this means that we are able to do transient analysis for any SPN for which we can construct the state space, provided we have enough memory to keep the vectors π. Using our IDDCSL prototype we are now able to compute CSL properties, which were formerly not amenable to analytic model checking, for examples see [GHL07], [HGD08]. We are working on improvements of the sketched optimisation techniques, in particular the parallelisation. Furthermore we are going to support nonordinary stochastic Petri nets and full CSL model checking. Thus, iterative solving of homogenous linear equation systems will be realised. For the time being we model capacities of places by introducing complementary places, which does its job, but blows up the models (e.g., our circadian clock model) and prohibits the use of predeﬁned functions as BioM assAction. To avoid such restrictions and eventually improve performance and memory consumption, we are going to support extended arc types, including the inhibitor arcs. There are other stochastic Petri net tools, oﬀering numeric analysis techniques as transient analysis which is the key to CSL model checking, e.g. SMART
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[CJMS06] and M¨obius [PCS07]. We will also compare our tool with them, including technical networks in a representative benchmark suite as well. Acknowledgements. We appreciate the support by the PRISM tool, which we use in our backtoback testing process as golden prototype. Snoopy’s export to PRISM has been implemented by Fei Liu, who is funded by the FMER (BMBF), funding number 0315449H. The export oﬀers various variable ordering options for comparison and teaching purposes.
References [ASSB00]
Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Model checking continuous time Markov chains. ACM Trans. on Computational Logic 1(1) (2000) [BHHK00] Baier, C., Haverkort, B., Hermanns, H., Katoen, J.P.: Model checking ContiuousTime Markov Chains by transient Analysis. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 358–372. Springer, Heidelberg (2000) [BL00] Barkai, N., Leibler, S.: Biological rhythms: Circadian clocks limited by noise. Nature 403, 267–268 (2000) [CDDS06] Cerotti, D., D’Aprile, D., Donatelli, S., Sproston, J.: Verifying stochastic wellformed nets with CSL model checking tools. In: Proc. ACSD 2006, pp. 143–152. IEEE Computer Society, Los Alamitos (2006) [CJMS06] Ciardo, G., Jones III, R.L., Miner, A.S., Siminiceanu, R.I.: Logic and stochastic modeling with smart. Perform. Eval. 63(6), 578–608 (2006) [CSK+ 03] Cho, K.H., Shin, S.Y., Kim, H.W., Wolkenhauer, O., McFerran, B., Kolch, W.: Mathematical modeling of the inﬂuence of RKIP on the ERK signaling pathway. In: Priami, C. (ed.) CMSB 2003. LNCS, vol. 2602, pp. 127–141. Springer, Heidelberg (2003) [CVOG05] Calder, M., Vyshemirsky, V., Orton, R., Gilbert, D.: Analysis of Signalling Pathways using the PRISM model checker. In: Proc. CMSB 2005, pp. 179– 190. LFCS, Univ. of Edinburgh (2005) [GH06] Gilbert, D., Heiner, M.: From Petri nets to diﬀerential equations  an integrative approach for biochemical network analysis. In: Donatelli, S., Thiagarajan, P.S. (eds.) ICATPN 2006. LNCS, vol. 4024, pp. 181–200. Springer, Heidelberg (2006) [GHL07] Gilbert, D., Heiner, M., Lehrack, S.: A unifying framework for modelling and analysing biochemical pathways using Petri nets. In: Calder, M., Gilmore, S. (eds.) CMSB 2007. LNCS (LNBI), vol. 4695, pp. 200–216. Springer, Heidelberg (2007) [HDG10] Heiner, M., Donaldson, R., Gilbert, D.: Petri Nets for Systems Biology. In: Iyengar, M.S. (ed.) Symbolic Systems Biology: Theory and Methods, Jones and Bartlett Publishers, Inc. (in press, 2010) [HGD08] Heiner, M., Gilbert, D., Donaldson, R.: Petri nets for systems and synthetic biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 215–264. Springer, Heidelberg (2008) [HRS08] Heiner, M., Richter, R., Schwarick, M.: Snoopy  a tool to design and animate/simulate graphbased formalisms. In: Proc. PNTAP 2008, associated to SIMUTools 2008. ACM digital library, New York (2008)
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Qualitative Transition Systems for the Abstraction and Comparison of Transient Behavior in Parametrized Dynamic Models Hayssam Soueidan1 , Gr´egoire Sutre2 , and Macha Nikolski2 1
LaBRI/ENSEiRB, Universit´e Bordeaux I, 351 crs Liberation, 33405 Talence, France CNRS/LaBRI, Universit´e Bordeaux I, 351 crs Liberation, 33405 Talence, France
2
Abstract. Quantitative models in Systems Biology depend on a large number of free parameters, whose values completely determine behavior of models. These parameters are often estimated by fitting the system to observed experimental measurements and data. The response of a model to parameter variation defines qualitative changes of the system’s behavior. The influence of a given parameter can be estimated by varying it in a certain range. Some of these ranges produce similar system dynamics, making it possible to define general trends for trajectories of the system (e.g. oscillating behavior) in such parameter ranges. Such trends can be seen as a qualitative description of the system’s dynamics within a parameter range. In this work, we define an automatabased formalism to formally describe the qualitative behavior of systems’ dynamics. Qualitative behaviors are represented by finite transition systems whose states contain predicate valuation and whose transitions are labeled by probabilistic delays. Biochemical system’ dynamics are automatically abstracted in terms of these qualitative transition systems by a random sampling of trajectories. Furthermore, we use graph theoretic tools to compare the resulting qualitative behaviors and to estimate those parameter ranges that yield similar behaviors. We validate this approach on published biochemical models and show that it enables rapid exploration of models’ behavior, that is estimation of parameter ranges with a given behavior of interest and identification of some bifurcation points.
1
Introduction
Dynamic models in System Biology rely on kinetic parameters to represent the range of possible behaviors when enzymatic information is incomplete. Analysis of these parametrized models aims at the identiﬁcation of parameter ranges yielding similar qualitative behaviors, or of parameter values yielding a given behavior of interest. Qualitative transient behavior can be successfully analyzed by model checking algorithms applied to models admitting a computable path semantics. However, in Systems Biology state explosion and negative decidability results limit the scope of model checking to a certain subset of models. Moreover, some published and curated Systems Biology models lack explicit semantics. Little can be assumed for these “black box” models, except the possibility of simulation. P. Degano and R. Gorrieri (Eds.): CMSB 2009, LNBI 5688, pp. 313–327, 2009. c SpringerVerlag Berlin Heidelberg 2009
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Mining these simulation results to identify parameter regions yielding similar behaviors is hindered by the size of the parameter space to explore, numerical artifacts and the lack of formal deﬁnition of what it means for simulation results to be similar. In this paper, we propose the new formalism of qualitative transition systems for abstracting simulation results in terms of discrete objects that admit eﬃcient similarity measures. Indeed, simulation results for ODEs are obtained using numerical integration schemes operating on ﬂoating point numbers. The resulting approximation is problematic for the identiﬁcation of precise transient properties since transient properties of interest are mathematically by equality between real numbers (e.g. f (x) = 0 is necessary for a local maximum) which is inconsistent in ﬂoating point arithmetic[1]. Consequently, even analysis of basic properties (like the detection of the ﬁrst time a deterministic system is in a previously visited state) fail in practice due to this inconsistency. Furthermore, diﬀerent integration schemes (nth order, implicit/explicit) yield diﬀerent and incomparable numerical approximations of the same trajectory. Although using normalized sampling and ﬁxed precision decimal numbers seem to solve this problem, the multiplicity of time scales in ODEs show that this solution is not completely satisfying. For dynamic models admitting a computable path semantics, the impact of numerical artifacts is absent. Indeed, it is possible to compute a ﬁnite description of the set of trajectories of the model. Consequently, for these models, model checking algorithms can decide if a logical representation of a behavior holds, and if not, can provide a counter example. Recently, a probabilistic model checking approach was successfully used to solve the inverse problem: given a logical representation of a transient behavior, return a parameter space in which any trajectory satisﬁes the speciﬁed behavior with suﬃciently high probability[2]. For dynamic models suitable for model checking, the intuitive notion of “similar behavior” is thus fully formalized and generally decidable. Contributions. In the next section we introduce Qualitative Transition Systems (QTS) and deﬁne their probabilistic semantics. A novel abstraction operation is deﬁned in section 3 with the goal of building QTSs from simulation results. We then show in section 4 that when constructing a QTS from an ODE, the QTS construction can be made independent of the numerical integration scheme. In section 5, we show that trajectory comparison using QTS can be made more resistant to noise by detecting points of interest (extremums and inﬂection) through the construction of a piecewise linear approximation (PLA). In section 6, we validate our approach on models from literature.
2
Qualitative Transition Systems
Given a set Σ, we denote by Σ ∗ the set of all (ﬁnite) words s0 · · · sk over Σ. A (ﬁnite) timed word over Σ is any word W = (t0 , s0 ) · · · (tk , sk ) ∈ (IR≥0 × Σ)∗ such that ti < ti+1 for i ∈ [0, k). The nonnegative real numbers ti are interpreted as the absolute observation times and the si are the observed values. We will
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focus in the paper on the particular case of Σ = IRn , where observed values are vectors of reals. In this case, timed words are called (multivariate) time series, and are denoted S = (t0 , x0 ) · · · (tk , xk ). We deﬁne a Qualitative Transition System (QTS) in the following way. Definition 1. A qualitative transition system is a tuple A = Q, E, μ, σ, w where Q is a ﬁnite set of set of qualitative states, E ⊆ Q × Q is a ﬁnite set of transitions, μ, σ : E → IR are mean and standard deviation labelings, and w : E → IN is a weight labeling. For any transition e ∈ E, μ(e) and σ(e) are respectively interpreted as being the mean and the standard deviation of a normal distribution that is followed by a random variable called sojourn time. The weight labeling w induces probabilities for transitions. Formally, the transition probability labeling p : E → [0, 1] induced by w is deﬁned by w(q, q ) p(q, q ) = . (q,q )∈E w(q, q ) A QTS is thus a transition system where each transition is labeled with the amount of time the system is idle before moving to another state. The delay between two state changes follows a parametrized normal distribution. This has to be contrasted with continuous Markov chains, where the sojourn time in a state must be exponentially distributed (see e.g. [3] for a complete deﬁnition). Suppose that a QTS is in the state q, and that there exists an outgoing transition e = (q, q ). The probability of moving from state q to state q is p(e), the transition probability of e. Suppose that the transition e is selected in favor of other outgoing transitions; the system will stay in the state q for a delay that is normally distributed with mean μ(e) and with standard deviation σ(e). Let X be such a normally distributed random variable that denotes the sojourn time in the state q, and let FX be its cumulative distribution function. The probability to move from q to q between t1 and t2 time units is thus given by FX (t2 ) − FX (t1 ). Contrary to the standard semantics of continuous time Markov chains, our semantics does not involve a race condition. That is, in a given state, the probability for the successor state is not conditioned by the delays but solely by the transition weights, similarly to a discrete time Markov Chain.
3 3.1
Abstraction of a Time Series in Terms of Qualitative Transition Systems Abstraction of a Time Series in Terms of Timed Words
In order to represent a realvalued trajectory as an abstractvalued trajectory, each concrete observation (t, x) of a time series S is transformed into an abstract observation (t, a) where the observation time t is unchanged and a is an abstract value in a ﬁnite domain A called the abstract domain. The rationale behind abstraction is that two concrete observations that are transformed into the same abstract observation are assumed indistinguishable w.r.t. qualitative properties.
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Formally, an abstraction function is any function α : IRn → A where A is a ﬁnite domain. For any time series S = (t0 , x0 ) · · · (tk , xk ) in (IR≥0 × IRn )∗ , the abstraction of S is the timed word α(S) = (t0 , α(x0 )) · · · (tk , α(xk )) in (IR≥0 × A)∗ . Note that abstraction functions may be combined by the cartesian product. In practice, it is often desirable to use multiplearity abstraction functions that are deﬁned on a ﬁxedwidth “window” of observations, i.e, functions (IRn )d+1 → A where d ∈ IN is the window width. For such a function α, the abstraction of S would be deﬁned as the timed word α(S) = (td , α(x0 , . . . , xd )) · · · (tk , α(xk−d , . . . , xk )). Observe that α(S) = α(S ) where S = (td , (x0 , . . . , xd )) · · · (tk , (xk−d , . . . , xk )) is a time series over IRnd . For simplicity, and without loss of generality, we only formalize our approach for unary abstraction functions (with zero width). For example, the sign function can abstract a time series into a timed word over the domain domain {−, 0, +}. The rank function can abstract a timed word over the domain {1, ..., n!} by mapping to each component of a vector its index in the corresponding sorted vector. For example, sort(11, −2, 1, 2) = (−2, 1, 2, 11) therefore rank(11, −2, 1, 2) = (4, 1, 2, 3). In the same way, the sign of the ﬁrst (resp. second) derivative can distinguish between intervals where the time series is increasing (resp. rapidly increasing) or decreasing (resp. rapidly decreasing). Abstracting the value of ﬁrst derivative (resp. second derivative) requires two points (resp. three points). x ≥ y, x ≥ 0, y ≤ 0 x ≤ y, x ≥ 0, y ≥ 0
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Fig. 1. Decomposition of a limit cycle of a two variables (x, y) system. By considering the sign and rank of the variables, we map an abstract value (here denoted by a formula) to each point of the trajectory. Boxes in the figure encompass successive points that are mapped to the same abstract value. Successive points of the trajectory are collapsed whenever they have the same abstract value, that is whenever they have the same sign and rank. This trajectory can thus be abstracted as a seven state QTS.
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Since the abstract domain is ﬁnite, it is often the case that the abstract time series α(S) has successive observations that are equal. These repeated observations are removed by collapsing them. Formally, for any timed word W = (t0 , a0 ) · · · (tk , ak ), let collapse(W ) be the timed word (ti0 , ai0 ) · · · (tih , aih ) where i0 < · · · < ih are such that i0 = 0 and aij = aij +1 = · · · = aij+1 −1 = aij+1 for every 0 ≤ j < h. Observe that collapsing is idempotent: for any timed word W , it holds that collapse(W ) = collapse(collapse(W )). The reduced abstraction of any time series S is then deﬁned as the timed word collapse(α(S)). 3.2
Abstraction of a Timed Word in Terms of Qualitative Transition System
The abstraction of a time series in terms of timed words abstracts the value component of the time series. In order to adequately compare two timed words, we also need to abstract the time of observations. Consider a timed word W = (t0 , a0 ) · · · (tk , ak ) over an abstract domain A. To qualitatively abstract this timed word, it is represented as a transition system by considering that for any i ∈ [0, k), the pair ((ti , ai ), (ti+1 , ai+1 )) of successive abstract observations of W is induced by a timed transition ai → ai+1 between two states of a transition system with a delay of ti+1 − ti . We can then consider the set of all transitions between two given states. From such a set of transitions with identical source and target, we suppose that the delays are approximately normal, and thus estimate the mean and standard deviation of the supposed underlying normal distribution. In this way, the set of concrete transitions can be abstracted by a single stochastic transition in a qualitative transition system. Formally, a timed word is abstracted in terms of QTS with the following deﬁnition. For any ﬁnite subset X ⊆ IR, we denote by E[X] the mean of X and by V[X] its variance. Definition 2. The QTS abstraction of a timed word W = (t0 , a0 ) · · · (tk , ak ) over A is the qualitative transition system A = Q, E, μ, σ, w with Q = {ai  0 ≤ i ≤ k} E = {(ai , ai+1 )  0 ≤ i < k ∧ ai = ai+1 }
μ(q, q ) = E[Δ(q, q )] σ(q, q ) = V[Δ(q, q )] w(q, q ) = Γ (q, q )
where for any (q, q ) ∈ E, Γ (q, q ) is the set of pairs (i, j) with 0 ≤ i < j ≤ k such that ai = q, aj = q , and ai−1 = ai = ai+1 = · · · = aj−1 , and Δ(q, q ) is the multiset deﬁned by Δ(q, q ) = {tj − ti  (i, j) ∈ Γ (q, q )}. Note that in the deﬁnition, the set Γ (q, q ) contains pairs of indices (i, j) such that all observations between i and j are removed by collapsing. Therefore, any two timed words W and W over A satisfying collapse(W ) = collapse(W ) have the same QTS abstraction.
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Abstraction of the Transient Behavior of Deterministic Parametrized Models
Deterministic parametrized models, such as ODE systems, can exhibit diﬀerent qualitative behaviors depending on the value of the parameters. When these
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systems admit a simulation algorithm (e.g. numerical integration), they generate time series. We show in this section that under assumptions concerning the simulation algorithms, the properties of interest of a given system are preserved by the abstraction in terms of qualitative transition systems. Sampling independence. In the context of time series obtained by sampling, the deﬁnition of QTS obviously depends on the precision of the sampling. However, we show that for convex abstraction functions if the sampling is “precise enough” then the QTS obtained from any oversampling has the same transitions (but with more precise delay distributions). We ﬁrst introduce additional notations. For any two vectors x, y ∈ IRn , we denote by x y the open line segment between x and y, formally x y = {λ x + (1 − λ) y  λ ∈ IR, 0 < λ < 1}. A time series S = (t0 , x0 ) · · · (tk , xk ) is a sampling of a (partial) function f : IR≥0 → IRn if xi = f (ti ) for every i ∈ [0, k]. Given a time series S = (t0 , x0 ) · · · (tk , xk ), we denote by ΛS : [t0 , tk ] → IRn its linear interpolation, deﬁned by: ΛS (ti ) = xi ti+1 −t t−ti for each 0 ≤ i ≤ k, and ΛS (t) = ti+1 −ti xi+1 + ti+1 −ti xi for each 0 ≤ i < k and ti < t < ti+1 . Given an abstraction function α : IRn → A, we say that α is convex if α−1 (a) is a convex subset of IRn for every a ∈ A. We consider for the remainder of this section a convex abstraction function α : IRn → A. When the sampling is precise enough, the linear interpolation ΛS is often used in practice, in place of the “real trajectory”. We formalize this notion of precision with respect to the abstraction function. A time series S = (t0 , x0 ) · · · (tk , xk ) is called αadequate if α(x) ∈ {α(xi ), α(xi+1 )} for every i ∈ [0, k) and x ∈ xi xi+1 . It is called αloose otherwise. Intuitively, when S is αadequate, the abstraction function α along xi xi+1 is either constant, or is ﬁrst equal to α(xi ) on xi z and then equal to α(xi+1 ) on z xi+1 , for some z ∈ xi xi+1 . Indeed, if α(xi xi+1 ) ⊆ {a, b} with a = α(xi ) and b = α(xi+1 ) being distinct, then, by convexity of α, the segment xi xi+1 is partitioned into the two convex subsegments xi xi+1 ∩ α−1 (a) and xi xi+1 ∩ α−1 (b), and z is at the boundary between these two subsegments. Therefore, an αadequate time series S captures all changes of α along its linear interpolation ΛS . However, these changes are captured up to the precision ti+1 − ti of the sampling, which leads us to the following deﬁnition. The αﬁtting of an αadequate time series S = (t0 , x0 ) · · · (tk , xk ) is the abstract timed word S = (tˆ0 , a ˆ0 ) · · · (tˆk , a ˆk ) deﬁned ˆ by a ˆi = α(xi ), t0 = t0 , and ti+1 if α(xi ) = α(xi+1 ) tˆi+1 = inf {t  t ≥ ti ∧ α(ΛS (t)) = α(xi+1 )} otherwise. Proposition 1. For any αadequate time series S = (t0 , x0 ) · · · (tk , xk ), the E, μ respective QTS abstractions Q, E, μ, σ, w and Q, , σ , w of α(S) and S satisfy Q = Q, E = E, w = w. Moreover, letting Δ = max ti − tˆi  0 ≤ i ≤ k , μ(e) − μ (e) ≤ Δ and σ 2 (e) ≤ σ 2 (e) + 4Δ2 + 4Δ σ (e) w(e) for every e ∈ E. Note that in the above proposition, we have Δ ≤ max {ti+1 − ti  0 ≤ i < n}. Hence, the error on μ (w.r.t. to the αﬁtted one) is bounded by the sampling period.
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An oversampling of S is any sampling U = (u0 , y0 ) · · · (ul , yl ) of ΛS such that t0 · · · tk is a subsequence of u0 · · · ul . It follows from the deﬁnitions that any oversampling of an αadequate time series S is also αadequate. Note that, informally, the αﬁtting of an oversampling of S is an “abstract oversampling” of the αﬁtting of S. According to Proposition 1, the QTS obtained by oversampling S is equal to the QTS obtained from S, except for the imprecision on μ and σ. This shows that for qualitative analysis there is little to be gained by oversampling: αadequate time series are suﬃcient. Periodic orbits detection. Oscillations are ubiquitous qualitative behaviors found in systems with a feedback loop. Although bifurcation analysis provides numerical methods to establish the presence of periodic orbits for ODEs, these methods cannot be applied to a general deterministic system such as an ODE with events. However, we show in this section that a QTS can be used eﬃciently to estimate the likelihood of a periodic orbit in a time series. Under an adequate abstraction function, a QTS that abstracts the transient behavior of a system with a periodic orbit has cycles in its transition relation. Consider a QTS obtained by applying the abstraction function α to an αadequate time series S obtained by sampling a continuous function f : t → IRn . By deﬁnition, f admits an orbit if and only if there exists a time point t and a period π such that f (t) = f (t + π). Furthermore, f admits a periodic and non constant orbit if and only if there exists an intermediate time step t < t + π such that f (t ) = f (t + π). Since S is adequately sampled for α, there exist at least three successive diﬀerent values in collapse(α(S)) and consequently the resulting QTS has at least a cycle of length 1. Since equality between the real numbers and their ﬂoating point approximation is not coherent, detection of periodic orbits for a time series must rely on estimations. To ﬁnd a periodic orbit in a time series S it is suﬃcient to ﬁnd a period π ∈ IR≥0 such that there exist two elements (ti , xi ), (tj , xj ) ∈ S such that (tj , xj ) ≈ (ti + π, xi ) for an adequate approximation relation ≈. However, for ODE systems integrated with an adaptive time step algorithm this scheme produces mainly false positives (successive integration steps in a quasi steady region of an ODE) and false negatives (regions with high variability). The existence of a periodic orbit of period π also implies that for any value k ∈ IN, f (t) = f (t + k ∗ π). Thus, if the system reaches a periodic orbit at point l, then the nearest points (according to an euclidean distance on IRn ) of l contain points from all possible periods. Therefore, we estimate the likelihood of a periodic orbit by considering a point l = (tl , xl ) of S that we suppose being in the periodic orbit, and a set of sample points P from S such that for any point p = (tp , xp ) in S − P , for any point p = (tp , xp ) in P , we have  xp − xl > xp − xl . Less formally, P is a set containing the points that are the nearest to l w.r.t. the euclidean distance. The likelihood L((tl , xl ), π, P ) of π being the period of the orbit of xl given a sample P of neighbors of xl is then deﬁned by
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L((tl , xl ), π, P ) =
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5
Accounting for Noise by Comparing Critical Points
Qualitative transition systems can capture the dynamics of a time series, even if the time series contains numerical errors that are only local. In the case of time series admitting global noise, abstraction functions that were adequate for a smooth time series may not be resistant to noise and can generate a QTS that inadequately captures the dynamics of noise. For example, abstracting with the sign of the ﬁrst derivate can adequately detect oscillations[2] but fails for time series even with little noise. Although moving average can smooth a time series and seem to circumvent this problem, the size of the window must be ﬁxed a priori and this approach is thus neither general nor adaptive. We propose here an adaptive approach to capture the most important points w.r.t. the shape of a time series. The critical points of continuous function f : IR → IR are the set of points where f (x) = 0. These are points where the function f either has a peak and changes direction (local or global extremum) or presents a curvature change (inﬂection points). In both cases the shape of f changes around the point. We generalize this deﬁnition to time series in the following way. Definition 3. The critical point of a time series S = (t0 , x0 ) · · · (tk , xk ) is the point (tc , xc ) ∈ S maximizing the function Λ(tc , xc ) = xc − x0  + (tc − t0 ) ∗ (xk − x0 )/(tk − t0 ). The critical point of a time series is the point of maximal distance with the linear interpolation between the ﬁrst and last points of the series. In a numerical context, this point is uniquely deﬁned. A critical point splits the time series in two time series. Since a critical point is also deﬁned for these series, we can recursively approximate a time series by considering a piecewise function which is linear between critical points. Definition 4. The piecewise linear approximation of order i (hereafter PLA) of a time series is the piecewise linear function on the intervals I0 , ..., Ik where any interval Ij has a lower bound (resp. upper bound) corresponding to the location of the j th (resp. j + 1) critical point. In order to compute the PLA of a time series, we deﬁne the piecewise linear interpolations of a set of points as the union of the linear interpolation between
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two successive points. The computation of the PLA of order i is then performed as follows. PLA(S, i) returns a list of critical points of S = (t0 , x0 ) · · · (tk , xk ) for each dimension of S. 1. For each dimension d ∈ [0, dim(x)], (a) Initialize the critical points with the ﬁrst and the last point of S projected on the dimension d : Cd ← {(t0 , πd (x0 )), (tk , πd (xk ))}. (b) While  Cd < i, i. Compute the linear interpolation between each successive pair of Cd : t−tj tj+1 −t Let Λj (t) = tj+1 −t xj+1 + tj+1 −tj xj ,where (tj , xj ) and (tj+1 , xj+1 ) j are two successive points in Cd ii. Build the piecewise linear interpolation: let Λ(t) = Λj (t) for t ∈ [tj , tj+1 ], iii. Let (tc , xc ) = argmax {πd (xc ) − Λ(tc )  (tc , xc ) ∈ S}, iv. Insert (tc , πd (xc )) in Cd s.t. Cd remains sorted w.r.t. the ﬁrst component 2. Return {Cd  d ∈ [0, dim(x)]} The previous algorithm cannot append a point twice to the list of critical points. Indeed, once a point is appended to the list it becomes a bound of the piecewise linear interpolation that is used for determining the next critical point. Consequently, at this point the distance between the next piecewise linear interpolation and the time series is 0, and the distance can not be maximized. Note that this does not hold for the piecewise linear regression. Which implies that, for any unidimensional time series, the segmented linear regression of i intervals 6
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minimizing the residuals with the time series can be obtained by considering the critical points as bounds of the interval. Examples of critical points. Critical points are highly related to the shape of the time series. Consider for example the sigmoid shape: a simple shape descriptor may simply specify that we start from a low plateau, follow an almost vertical increase before reaching another high plateau. Such a sigmoid shape exhibits two critical points, one at the end of the ﬁrst (low) plateau, and one at the start of the second (high) plateau. Similarly, consider an oscillatory time series such as the one depicted in ﬁgure 2: its critical points contain the successive highest local maximum and the lowest local minimum.
6
Case Studies and Experimental Results
In this section we show how our approach can be used in practice by solving four problems related to qualitative behavior analysis. Although each problem and solution is illustrated on a speciﬁc model, the methods that are used are generalpurpose. All the models used in this section were downloaded from the BioModels database[4] in the SBML V2 L1 format[5], simulated using MathSBML[6] and were used without any modiﬁcation. Simulations were performed on an Intel Core2 3,2GH personal computer and each algorithm was allowed to run for at most ﬁve minute. If parameter values are not speciﬁed in the case studies, it means that those provided in the SBML ﬁle were used. 6.1
Searching a Trajectory with a Given Periodic Orbit
The ﬁrst model we consider is a model of the cell cycle based on the interactions between the cyclin dependent kinase cdc2 and cyclin [7]. The model is comprised of six variables and ten parameters. We consider the following problem. Given the representative trajectory and its associated parameters described in the original article (left in ﬁgure 3), what kind of similar trajectories can be found in the whole parameter space ? Abstracting the behavior of the left ﬁgure with a rank abstraction function yields the QTS depicted in the right part of ﬁgure 3. Notice the highlighted non deterministic states. These states and transitions are due to numerical errors and happen while the system reaches its periodic orbit. Consequently, the weights of the outgoing highlighted transitions are 1 while the incoming transitions are 17. All other transitions in the single cycle of the QTS have a weight of 18. We obtained 500 random samples for the six parameters considered as being critical by the original author. For each parameter sample, we computed the trajectory, abstracted it in terms of QTS by applying the rank function and computed the Sorensen similarity index over the set of transitions to compare the sampled QTS with the representative QTS. Figure 4 depicts a subset of the results. Note that we chose parameter values exhibiting “similar” sustained oscillations, but of diﬀerent transient behaviors. We then compared these results
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with the one obtained with a stochastic simulation algorithm. For each simulation result, we used the PLA algorithm to reduce each noisy trajectory to its 50 most critical points, and abstracted these points in terms of QTS by applying the rank function. Trajectories similar to the one simulated with numerical integration were found for comparable parameters values.
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Fig. 5. Oscillation period for the MAPK cascade model. Left: contour plot representing the period of oscillations. The bottom axis (resp. left axis) represents values of the k4 (resp. v5) parameter. The contour plot was built with 500 simulations with random parameters. Regions with comparable periods are represented by a region with uniform color. Right: Two example trajectories exhibiting oscillations of minimal and maximal period A:1094 time units and B:2236 time units.
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Estimating the Period of Orbits
The model of MAPK cascade from [8] describes the eﬀect of negative feedback and ultrasensitivity on the emergence of oscillations. We investigated the dynamics of the period of the orbits under parameter changes. To estimate these periods, we considered the parameters {k4 , v5 } as random variables following an uniform distribution over the intervals [0, 1] and [0, 0.1]. For parameters’ sample of size 500, we abstracted the corresponding time series in terms of QTS. These QTS were then reduced by removing transitions whose probability decreased as the simulation advanced. We then approximated the orbit’s period with the sum of means of the transitions of the longest cycle of the QTS. This approximation was then used in a local maximization procedure to identify the exact period value maximizing the likelihood function. In our tests, providing this initial “educated guess” of the period value to the maximization procedure yielded the global maximum in 98% of cases. We can see from the results (ﬁgure 5) that, for this parameter subspace, oscillating behavior is very common and that the dynamics of the period does not exhibit abrupt changes. 6.3
Searching for Any Periodic Orbits
We consider again the MAPK cascade model [8] but with a more general objective. We consider the problem of detecting the possible oscillating behaviors and
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Fig. 6. Example trajectories of the MAPK cascade exhibiting oscillating behavior found with a random sampling of ten parameters of the MAPK cascade model [8]. On the right of each plot, the associated QTS reduced to its periodic form; under it, the period value with maximum likelihood.
of computing the probability of ﬁnding an oscillating behavior in a larger parameter subspace. The parameters of interest are {k3 , k4 , k7 , k8 , V5 , V6 , V9 , V10 } and are considered as random variables following an uniform probability over the interval [0, 1] ⊂ IR. We built a QTS as in previous sections. In this study, only periods in the range [200, 5000] with a likelihood greater than 10 were considered genuine. Although all the resulting trajectories exhibit transient or limit cycle oscillations, they follow diﬀerent transient dynamics. The four example trajectories of ﬁgure 6 show a subset of the possible dynamics: each of these time series admits a speciﬁc alternation of species at their maximum concentration. Multiple instances of each of these dynamics were successfully identiﬁed by applying the method from section 6.1. The number of samples needed before ﬁnding an oscillating behavior was 57 on average. For comparison, when k3 , k4 , k7 and k8 were sampled in the interval [0, 0.1], the average number of samples needed dropped to 10.6. 6.4
Searching for Given Transient Behavior in a Parameter Subspace
The extracellular signal regulated kinase (ERK) pathways plays a role in a hidden oncogenic positive feedback loop via a crosstalk with the Wnt pathway [9]. The pathological cases identiﬁed by the authors involve “an irreversible response leading to a sustained activation of both pathways”. Applying our QTS construction with random samples of the βcatenin synthetic rate (V12 ) yields results depicted in ﬁgure 7. This model involves 28 species, 58 parameters, and 2 discrete events. Applying the rank abstraction yields transition systems with a state space of 600 states on average out of the possible 28! state conﬁgurations.
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Discussion and Conclusions
In this paper, we have described the formalism of qualitative transition systems. A QTS is a transition system where each transition is labeled with the amount of time the system requires before moving to another state. The delay between two state changes follows a parametrized normal distribution. We have shown how QTS can be used to study qualitative properties of parametrized models. This is achieved by deﬁning an appropriate abstraction function. By representing the characteristic qualitative features of a trajectory in an abstract domain that is countable, qualitative similarity can be detected by a simple equality test. We have shown that the soundness of this approach depends on the adequacy of sampling with respect to the abstraction function. In particular, we have shown that for convex abstraction functions, if the sampling is “precise enough”, then the QTS obtained from any oversampling has the same transitions. Finally, we applied this approach to some well known models. QTS were used to explore the parameter space and to detect uniform behaviors (oscillations etc.). The limits of our approach as compared to model checking is the lack of exhaustivity. This has to be counterbalanced by the fact that our method is applicable to a large panel of formalisms, even those lacking a precise semantics. Consequently, we can avoid any model transformation. Finally, our approach can be applied independently to the data and to the model. The areas of future research for qualitative transition systems can be declined on the both technical
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and practical plans. As for the former, we envision a more thorough study of similarity measures of QTS and how QTS similarity relates to language equivalence. As for the latter, we plan to develop clustering techniques in order to detect the resulting behavior similarity in an experimental context. Funding. This work was supported in part by the European Commission FP6 programme “Yeast Systems Biology Network” (YSBN), LSHGCT2005018942.
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Author Index
Akman, Ozgur E.
52
Back, RalphJohan 111 Baggi, Michele 68 BaillyBechet, Marc 83 Balbo, Gianfranco 281 Ballarini, Paolo 26 Ballis, Demis 68 Bartoli, Lisa 1 Bella, Giampaolo 96 Braunstein, Alfredo 83 Bussolino, Federico 281 Casadio, Rita 1 Chaouiya, Claudine 266 Ciocchetta, Federica 52 Clarke, Edmund M. 218 Cordero, Francesca 281 Czeizler, Elena 111 Czeizler, Eugen 111 Dang, Thao 126 De Maria, Elisabetta 142 Degasperi, Andrea 52 Delzanno, Giorgio 158 De Pierro, Massimiliano 281 Didier, Fr´ed´eric 173 Di Giusto, Cinzia 158 Fages, Fran¸cois 142 Falaschi, Moreno 68 Fariselli, Piero 1 Gabbrielli, Maurizio 158 Galpin, Vashti 189 Gennemark, Peter 205 Guerriero, Maria Luisa 52 Heath, John K. 18 Heiner, Monika 296 Henzinger, Thomas A. 173 Hillston, Jane 189 Horv´ ath, Andr´ as 281 Hsu, David 251 Jha, Sumit K. John, Mathias
218 235
Laneve, Cosimo 158 Langmead, Christopher J. Legay, Axel 218 Le Guernic, Colas 126 Lhoussaine, C´edric 235 Li` o, Pietro 96 Liu, Bing 251 Maler, Oded 126 Manini, Daniele 281 Martelli, Pier Luigi 1 Mateescu, Maria 173 Naldi, Aur´elien 266 Napione, Lucia 281 Niehren, Joachim 235 Nikolski, Macha 313 Pavan, Simona 281 Petre, Ion 111 Picco, Andrea 281 Platzer, Andr´e 218 Priami, Corrado 26 Quaglia, Paola
26
Remy, Elisabeth Rossi, Ivan 1
266
Schwarick, Martin 296 Sereno, Matteo 281 Soliman, Sylvain 142 Soueidan, Hayssam 313 Sutre, Gr´egoire 313 Thiagarajan, P.S. 251 Thieﬀry, Denis 266 Veglio, Andrea Wedelin, Dag Wolf, Verena
281 205 173
Zavattaro, Gianluigi 158 Zecchina, Riccardo 83 Zuliani, Paolo 218
218